Exponents

When a number is multiplied by itself several times, the product can be written in a short way. Instead of ${2\cdot2\cdot2\cdot2\cdot2}$ we write $2^5$ and read it “2 to the fifth power”. The number that is multiplied, here $2,$ is the base. The small raised number, here $5,$ is the exponent: it tells how many equal factors there are. The whole expression $2^5$ is called a power, and its value is ${2^5=32}.$

Exponents save a lot of writing: a product of 45 factors, each equal to $a,$ is simply $a^{45}.$ Finding the value of a power is called raising to a power.

On this page: Definition · Negative bases · Laws of exponents · Zero exponent · Negative exponents · Fractional exponents · Simplifying · Exponential function · Comparing powers · Equations · Practice

Definition of a Power

The exponents $2$ and $3$ have their own names: $a^2$ is read “$a$ squared” and $a^3$ is read “$a$ cubed”. The names come from geometry. A square with the side $a$ has the area $a^2,$ and a cube with the edge $a$ has the volume $a^3.$

Example 1. Calculate the powers.

• ${5^3=5\cdot5\cdot5=125};$
• ${3^4=3\cdot3\cdot3\cdot3=81};$
• ${16^1=16};$
• ${10^3}={10\cdot10\cdot10}={1000\text{:}}$ in general, $10^n$ is written as $1$ followed by $n$ zeros;
• ${\left(\frac{2}{3}\right)^2=\frac{2}{3}\cdot\frac{2}{3}=\frac{4}{9}};$
• ${0.5^2=0.5\cdot0.5=0.25};$
• ${1^{100}=1}$ and ${0^7=0}\text{:}$ every power of $1$ is $1,$ and every power of $0$ with a natural exponent is $0.$

Exponents and the Order of Operations

An exponent belongs only to the number or letter written right in front of it. When it belongs to a whole expression, that expression is put in brackets. Powers are calculated after the operations in brackets, but before multiplication, division, addition and subtraction.

Example 2.

• ${2\cdot3^2=2\cdot9=18},$ but ${(2\cdot3)^2=6^2=36};$
• ${5+2^3=5+8=13},$ but ${(5+2)^3=7^3=343};$
• $3x^2$ means ${3\cdot x\cdot x}\text{:}$ only $x$ is squared, while ${(3x)^2=3x\cdot3x=9x^2}.$

Powers of Negative Numbers

A product of negative numbers is positive when the number of factors is even, and negative when it is odd. So:

Example 3.

• ${(-2)^2}={(-2)\cdot(-2)}={4;}$
• ${(-2)^3}={(-2)\cdot(-2)\cdot(-2)}={-8;}$
• ${(-2)^4=16}$ and ${(-2)^5=-32};$
• ${(-1)^{100}=1}$ and ${(-1)^{101}=-1}.$

In general, ${(-a)^n=a^n}$ when $n$ is even, and ${(-a)^n=-a^n}$ when $n$ is odd. An even power of a number is never negative: for example, ${a^2\ge0}$ for every $a.$

Watch the brackets: ${(-3)^2}={(-3)\cdot(-3)}={9,}$ but ${-3^2=-(3\cdot3)=-9}.$ In $-3^2$ the exponent belongs only to $3,$ and the minus sign is applied after the power.

Laws of Exponents

Five rules make calculations with powers easy. Each of them follows from counting the factors. Here we prove them for natural exponents; later in this lesson we extend them to zero, negative and fractional exponents.

Multiplying Powers with the Same Base

${2^3\cdot2^4}={(2\cdot2\cdot2)}\cdot {(2\cdot2\cdot2\cdot2)}={2^7\text{:}}$ the first power has $3$ factors equal to $2,$ the second one has $4,$ so together there are ${3+4=7}$ factors. In the same way, $a^m$ has $m$ factors equal to $a$ and $a^n$ has $n$ of them, so their product has ${m+n}$ factors.

Example 4. ${2^6\cdot2^3=2^{6+3}=2^9=512};$ ${x^5\cdot x=x^{5+1}=x^6};$ ${a^2\cdot a^3\cdot a^4=a^{2+3+4}=a^9}.$

Dividing Powers with the Same Base

When we divide $5^6$ by $5^4,$ four factors $5$ in the numerator cancel with the four factors in the denominator:

${\frac{5^6}{5^4}}={\frac{5\cdot5\cdot5\cdot5\cdot5\cdot5}{5\cdot5\cdot5\cdot5}}={5\cdot5}={5^2}$

Two factors are left, and ${2=6-4}.$ In general, if ${a\ne0}$ and ${m\gt n},$ cancelling $n$ factors leaves ${m-n}$ factors.

Example 5. ${\frac{15^{10}}{15^8}}={15^{10-8}}={15^2}={225;}$ ${\frac{7^{12}}{7^{11}}=7^1=7};$ ${\frac{x^9}{x^4}=x^5}$ for ${x\ne0}.$

The base cannot be zero, because we cannot divide by zero. What happens when ${m=n}$ or ${m\lt n}$ is explained in the sections Zero Exponent and Negative Exponents.

Raising a Power to a Power

${(3^2)^3}={3^2\cdot3^2\cdot3^2}={3^{2+2+2}}={3^6\text{:}}$ there are $3$ groups with $2$ factors equal to $3$ in each, so ${2\cdot3=6}$ factors. In general, $(a^m)^n$ has $n$ groups of $m$ factors.

Example 6. ${(5^3)^4=5^{3\cdot4}=5^{12}};$ ${(x^2)^5=x^{10}};$ ${\left((-2)^2\right)^3=(-2)^6=64}.$

Do not confuse $(a^m)^n$ with ${a^m\cdot a^n}\text{:}$ ${(2^3)^2=2^6=64},$ but ${2^3\cdot2^2=2^5=32}.$ A “tower” of exponents is different: it is calculated from the top down, so ${2^{3^2}=2^9=512}.$

The Power of a Product

${(2\cdot5)^3}={(2\cdot5)}\cdot {(2\cdot5)\cdot(2\cdot5)}={(2\cdot2\cdot2)\cdot(5\cdot5\cdot5)}={2^3\cdot5^3\text{:}}$ we change the order of the factors and group the equal ones. Both sides are equal to $1000,$ because ${10^3=1000}$ and ${8\cdot125=1000}.$

The rule also works from right to left: to multiply powers with the same exponent, multiply the bases and keep the exponent.

Example 7.

• ${(2\cdot5)^2}={2^2\cdot5^2}={4\cdot25}={100;}$
• ${(3x)^4=3^4x^4=81x^4};$
• ${(-2a^3)^3}={(-2)^3(a^3)^3}={-8a^9;}$
• ${2^{12}\cdot3^{12}}={(2\cdot3)^{12}}={6^{12};}$
• ${4^5\cdot25^5}={(4\cdot25)^5}={100^5}={10^{10}.}$

The rule holds for three or more factors too, and together with the previous rule it gives ${(a^nb^m)^p=a^{np}b^{mp}}.$ For example, ${(x^2y^3)^4=x^8y^{12}}.$

The Power of a Quotient

${\left(\frac{2}{3}\right)^3}={\frac{2}{3}\cdot\frac{2}{3}\cdot\frac{2}{3}}={\frac{2\cdot2\cdot2}{3\cdot3\cdot3}}={\frac{2^3}{3^3}}={\frac{8}{27}\text{:}}$ when we multiply fractions, we multiply their numerators and their denominators.

Example 8. ${\left(\frac{1}{2}\right)^5=\frac{1}{32}};$ ${\left(-\frac{3}{4}\right)^2=\frac{9}{16}};$ ${\left(\frac{x}{3}\right)^3=\frac{x^3}{27}};$ ${\frac{15^4}{5^4}=\left(\frac{15}{5}\right)^4=3^4=81}.$

Here are the five laws together. They hold for any numbers $a$ and $b$ and natural numbers $m$ and $n,$ with ${a\ne0}$ and ${m\gt n}$ in the quotient rule and ${b\ne0}$ in the last rule.

Law	Example
${a^m\cdot a^n=a^{m+n}}$	${2^3\cdot2^4=2^7=128}$
${\frac{a^m}{a^n}=a^{m-n}}$	${\frac{5^6}{5^4}=5^2=25}$
${(a^m)^n=a^{mn}}$	${(3^2)^3=3^6=729}$
${(ab)^n=a^nb^n}$	${(2\cdot5)^3=2^3\cdot5^3=1000}$
${\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}}$	${\left(\frac{2}{3}\right)^3=\frac{2^3}{3^3}=\frac{8}{27}}$

There are no such rules for sums and differences. ${2^3+2^4=8+16=24},$ which is not ${2^7=128}.$ And $(a+b)^2$ is not equal to ${a^2+b^2}\text{:}$ ${(1+2)^2=9},$ but ${1^2+2^2=5}.$ Powers with different bases and different exponents cannot be combined by these laws either: ${2^3\cdot3^2=8\cdot9=72},$ not $6^5.$ When the bases are powers of one number, they can be combined, as in Example 15 below. The square of a sum is explained in the lesson Polynomial Identities.

Zero Exponent

What should $5^0$ mean? A product with zero factors makes no sense, so we choose the meaning that keeps the laws of exponents true. By the quotient rule, ${\frac{5^3}{5^3}=5^{3-3}=5^0}.$ But a number other than zero divided by itself is $1.$ So we define:

Example 9. ${76^0=1};$ ${(-5)^0=1};$ ${\left(\frac{2}{7}\right)^0=1};$ ${(x^2+1)^0=1},$ because ${x^2+1}$ is never zero. But ${-5^0=-1},$ because the exponent belongs only to $5.$

In school mathematics $0^0$ is left undefined: the argument above does not work for the base $0,$ because it would divide $0^3$ by $0^3,$ that is, divide by zero.

Negative Exponents

Look at the powers of $2.$ Each time the exponent goes down by $1,$ the value is divided by $2\text{:}$

$n$	$3$	$2$	$1$	$0$	$-1$	$-2$	$-3$
$2^n$	$8$	$4$	$2$	$1$	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{8}$

Going on past $0$ gives ${2^{-1}=\frac{1}{2}},$ ${2^{-2}=\frac{1}{4}=\frac{1}{2^2}}$ and ${2^{-3}=\frac{1}{8}=\frac{1}{2^3}}.$ The laws of exponents lead to the same meaning: ${a^{-n}\cdot a^n=a^{-n+n}=a^0=1},$ so $a^{-n}$ is the reciprocal of $a^n.$

A negative exponent does not make the number negative; it means $1$ divided by the power. For example, ${2^{-3}=\frac{1}{8}},$ not $-8$ and not $-6.$

Example 10.

• ${20^{-1}=\frac{1}{20}};$
• ${2^{-4}=\frac{1}{2^4}=\frac{1}{16}};$
• ${(-3)^{-2}=\frac{1}{(-3)^2}=\frac{1}{9}};$
• ${10^{-3}=\frac{1}{1000}=0.001};$
• ${\left(\frac{2}{3}\right)^{-2}=\left(\frac{3}{2}\right)^2=\frac{9}{4}}\text{:}$ a fraction with a negative exponent is turned upside down: ${\left(\frac{a}{b}\right)^{-n}=\left(\frac{b}{a}\right)^n}$ for ${a\ne0}$ and ${b\ne0};$
• ${\frac{1}{x^{-3}}=\frac{1}{\frac{1}{x^3}}=x^3}$ for ${x\ne0}\text{:}$ a factor can move from the denominator to the numerator (or back) if the sign of its exponent changes.

With zero and negative exponents, all five laws of exponents hold for all integers $m$ and $n,$ as long as the bases are not zero. Now the quotient rule works for any two exponents: ${\frac{a^3}{a^5}=a^{3-5}=a^{-2}=\frac{1}{a^2}}.$ Cancelling gives the same answer, because ${\frac{a\cdot a\cdot a}{a\cdot a\cdot a\cdot a\cdot a}=\frac{1}{a\cdot a}}.$

Example 11. Simplify and write the answer without negative exponents.

${x^5\cdot x^{-8}}={x^{5+(-8)}}={x^{-3}}={\frac{1}{x^3}}$

${(2^{-3})^{-2}}={2^{(-3)\cdot(-2)}}={2^6}={64}$

${\frac{a^{-2}b^3}{a^4b^{-1}}}={a^{-2-4}b^{3-(-1)}}={a^{-6}b^4}={\frac{b^4}{a^6}}$

$3x^{-2}=3\cdot\frac{1}{x^2}=\frac{3}{x^2}$ the exponent belongs only to $x$

Powers of $10$ with negative exponents are used to write small numbers: ${10^{-1}=0.1},$ ${10^{-2}=0.01},$ ${10^{-3}=0.001}.$ In $10^{-n},$ the digit $1$ stands in the $n$-th place after the decimal point. For example, a millimeter is $10^{-3}$ of a meter.

Fractional Exponents

An exponent can also be a fraction. Again we choose the meaning that keeps the laws of exponents true. By the power-of-a-power rule, ${\left(9^{\frac{1}{2}}\right)^2=9^{\frac{1}{2}\cdot2}=9^1=9}.$ So the square of $9^{\frac{1}{2}}$ must be $9.$ Two numbers have this property, $3$ and $-3;$ we take the positive one, so that powers of a positive number stay positive: ${9^{\frac{1}{2}}=\sqrt{9}=3}.$ In the same way, ${\left(8^{\frac{1}{3}}\right)^3=8},$ so ${8^{\frac{1}{3}}=\sqrt[3]{8}=2}.$

Here $\sqrt[n]{a},$ the $n$-th root of ${a\gt0},$ is the positive number whose $n$-th power is $a\text{:}$ for example, ${\sqrt[4]{16}=2},$ because ${2^4=16}.$ In $a^{\frac{m}{n}},$ the denominator $n$ is the index of the root, and the numerator $m$ is the exponent of the power. When you calculate, it is usually easier to take the root first, because the numbers stay small.

Example 12.

• ${16^{\frac{1}{2}}=\sqrt{16}=4};$
• ${27^{\frac{1}{3}}=\sqrt[3]{27}=3};$
• ${14^{\frac{1}{3}}=\sqrt[3]{14}}\text{:}$ this root is not a whole number;
• ${8^{\frac{2}{3}}=\left(\sqrt[3]{8}\right)^2=2^2=4};$
• ${16^{\frac{3}{4}}=\left(\sqrt[4]{16}\right)^3=2^3=8};$
• ${x^{\frac{3}{4}}=\sqrt[4]{x^3}}$ for ${x\gt0};$
• ${32^{-\frac{2}{5}}}={\frac{1}{32^{\frac{2}{5}}}}={\frac{1}{\left(\sqrt[5]{32}\right)^2}}={\frac{1}{4};}$
• ${0.25^{\frac{1}{2}}=\sqrt{0.25}=0.5}.$

Fractional powers are defined only for a positive base (and for the base $0$ with a positive exponent: ${0^{\frac{m}{n}}=0}$). With a negative base the rules would contradict each other. For example, ${\frac{1}{3}=\frac{2}{6}},$ but ${\sqrt[3]{-8}=-2},$ while ${\sqrt[6]{(-8)^2}=\sqrt[6]{64}=2}.$ That is why we use fractional powers only with positive bases. Some books and calculators give $(-8)^{\frac{1}{3}}$ the value $-2,$ but then the laws of exponents can fail. The cube root of $-8$ is written with the root sign: ${\sqrt[3]{-8}=-2}.$

For positive bases, the laws of exponents hold for fractional exponents too, and they are often the shortest way to work with roots: ${\sqrt{a}\cdot\sqrt[3]{a}}={a^{\frac{1}{2}}\cdot a^{\frac{1}{3}}}={a^{\frac{1}{2}+\frac{1}{3}}}={a^{\frac{5}{6}}}={\sqrt[6]{a^5}}$ for ${a\gt0}.$ You will find more examples in the lesson Radicals.

For ${a\gt0},$ the power $a^x$ can be defined for every real exponent $x,$ even an irrational one. The numbers $1.4,$ $1.41,$ $1.414,$ … get closer and closer to ${\sqrt{2}=1.41421\ldots},$ and the powers $2^{1.4},$ $2^{1.41},$ $2^{1.414},$ … get closer and closer to one number, which is $2^{\sqrt{2}}\text{:}$ ${2^{\sqrt{2}}\approx2.67}.$ For positive bases, the laws of exponents hold for all real exponents. With a negative base they can fail: ${\left((-3)^2\right)^{\frac{1}{2}}=\sqrt{9}=3},$ not $-3.$

Simplifying Expressions with Exponents

To simplify an expression, use the laws one step at a time: first remove the brackets with the power rules, then multiply or divide the powers with the same base. The numbers are multiplied as usual.

Example 13. Simplify ${(2x^3y)^2\cdot3xy^4}.$

${(2x^3y)^2\cdot3xy^4}={4x^6y^2\cdot3xy^4}$ the power of a product

$=12x^{6+1}y^{2+4}=12x^7y^6$ we multiply the numbers and add the exponents

Example 14. Simplify $\frac{(a^2b^3)^4}{a^5b^{10}}$ for ${a\ne0}$ and ${b\ne0}.$

${\frac{(a^2b^3)^4}{a^5b^{10}}}={\frac{a^8b^{12}}{a^5b^{10}}}={a^{8-5}b^{12-10}}={a^3b^2}$

Example 15. Calculate $\frac{4^5\cdot8^2}{16^3}$ without a calculator.

All the bases are powers of $2\text{:}$ ${4=2^2},$ ${8=2^3}$ and ${16=2^4}.$

${\frac{4^5\cdot8^2}{16^3}}={\frac{(2^2)^5\cdot(2^3)^2}{(2^4)^3}}={\frac{2^{10}\cdot2^6}{2^{12}}}={\frac{2^{16}}{2^{12}}}={2^4}={16}$

Example 16. Calculate ${\left(\frac{1}{3}\right)^{-2}+5^0-27^{\frac{2}{3}}}.$

${\left(\frac{1}{3}\right)^{-2}+5^0-27^{\frac{2}{3}}}={3^2+1-\left(\sqrt[3]{27}\right)^2}={9+1-9}={1}$

The Exponential Function

If the base ${a\gt0}$ stays fixed and the exponent changes, we get the exponential function ${y=a^x}.$ For ${a=1}$ it is the constant ${y=1},$ so we take ${a\ne1}.$ Here are the values of ${y=2^x}$ and ${y=\left(\frac{1}{2}\right)^x}$ for a few values of $x\text{:}$

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$2^x$	$\frac{1}{8}$	$\frac{1}{4}$	$\frac{1}{2}$	$1$	$2$	$4$	$8$
$\left(\frac{1}{2}\right)^x$	$8$	$4$	$2$	$1$	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{8}$

The graphs show the properties of every exponential function ${y=a^x}\text{:}$

• it is defined for every real number $x;$
• its values are always positive: the graph lies above the $x$-axis, gets closer and closer to it on one side, but never touches it;
• the graph passes through the point ${(0,\ 1)},$ because ${a^0=1};$
• for ${a\gt1}$ the function is increasing: the larger $x,$ the larger $a^x;$
• for ${0\lt a\lt1}$ the function is decreasing: the larger $x,$ the smaller $a^x.$

The two graphs are mirror images of each other in the $y$-axis, because ${\left(\frac{1}{2}\right)^x=2^{-x}}.$ You can change the base in the function graph plotter: move the slider $a$ and watch the graph. The inverse of the exponential function is the logarithm, explained in the lesson Logarithms.

Example 17. A colony of bacteria doubles every hour. If there are $100$ bacteria now, then after $t$ hours there will be ${100\cdot2^t}$ of them: after $1$ hour ${100\cdot2^1=200},$ after $3$ hours ${100\cdot2^3=800},$ and after $5$ hours ${100\cdot2^5=3200}.$ When a quantity is multiplied by the same number in equal steps of time, we speak of exponential growth; it soon becomes very fast.

Comparing Powers

Powers with the Same Base

Because the exponential function is increasing for ${a\gt1}$ and decreasing for ${0\lt a\lt1}\text{:}$

Example 18. ${4^2\lt4^3},$ because ${4\gt1}$ and ${2\lt3};$ indeed, ${16\lt64}.$ But ${\left(\frac{1}{2}\right)^2\gt\left(\frac{1}{2}\right)^3},$ because ${\frac{1}{4}\gt\frac{1}{8}}\text{:}$ multiplying by $\frac{1}{2}$ once more makes the number smaller. In the same way, ${3^{-5}\lt3^{-2}}$ and ${0.7^{10}\lt0.7^3}.$

Powers with the Same Exponent

Example 19. ${3^2\lt5^2},$ because ${9\lt25}.$ But ${3^{-2}\gt5^{-2}}\text{:}$ ${3^{-2}=\frac{1}{9}}$ and ${5^{-2}=\frac{1}{25}},$ and ${\frac{1}{9}\gt\frac{1}{25}},$ because of two fractions with the same numerator, the one with the smaller denominator is larger.

The bases must be positive here: ${-3\lt2},$ but ${(-3)^2=9}$ is greater than ${2^2=4}.$

Different Bases and Exponents

First write the two powers with the same base or with the same exponent.

Example 20. Which is larger, $27^5$ or $9^7?$ Both bases are powers of $3\text{:}$ ${27^5=(3^3)^5=3^{15}}$ and ${9^7=(3^2)^7=3^{14}}.$ Since ${15\gt14},$ we get ${27^5\gt9^7}.$

Example 21. Which is larger, $2^{30}$ or $3^{20}?$ The exponents have the common factor $10\text{:}$ ${2^{30}=(2^3)^{10}=8^{10}}$ and ${3^{20}=(3^2)^{10}=9^{10}}.$ Since ${8\lt9},$ we get ${2^{30}\lt3^{20}}.$

Equations with the Unknown in the Exponent

An exponential function is increasing or decreasing, so two powers with the same base ${a\gt0},$ ${a\ne1},$ are equal only when their exponents are equal: ${a^x=a^y}$ means ${x=y}.$ To solve an equation like ${2^x=32},$ write both sides as powers of the same base.

Example 22. Solve the equations: a) ${2^x=32};$ b) ${3^{x-1}=\frac{1}{9}};$ c) ${4^x=8}.$

a) ${32=2^5},$ so ${2^x=2^5}$ and ${x=5}.$

b) ${\frac{1}{9}=3^{-2}},$ so ${3^{x-1}=3^{-2}}.$ Then ${x-1=-2}$ and ${x=-1}.$

c) ${4^x=(2^2)^x=2^{2x}}$ and ${8=2^3},$ so ${2x=3}$ and ${x=\frac{3}{2}}.$ Check: ${4^{\frac{3}{2}}=\left(\sqrt{4}\right)^3=2^3=8}.$

An equation such as ${2^x=5}$ cannot be solved this way, because $2$ and $5$ are not powers of the same whole number. Since ${2^2=4}$ and ${2^3=8},$ the solution lies between $2$ and $3\text{:}$ it is the logarithm ${x=\log_2 5\approx2.32}.$

Practice Problems

1. Calculate: a) $3^4;$ b) $(-2)^5;$ c) $-2^4;$ d) $\left(\frac{2}{5}\right)^3;$ e) $0.1^3;$ f) $(-1)^{25}.$

Answer

a) ${3^4=81}.$ b) ${(-2)^5=-32}.$ c) ${-2^4}={-(2\cdot2\cdot2\cdot2)}={-16.}$ d) ${\left(\frac{2}{5}\right)^3=\frac{8}{125}}.$ e) ${0.1^3=0.001}.$ f) ${(-1)^{25}=-1}.$

2. Write as one power: a) ${a^4\cdot a^7};$ b) $\frac{x^{12}}{x^5};$ c) $(y^3)^6;$ d) ${5^3\cdot5^{-5}};$ e) $(2^{-2})^{-3};$ f) $\frac{3^{-4}}{3^2}.$

Answer

a) $a^{11}.$ b) $x^7.$ c) $y^{18}.$ d) ${5^{3+(-5)}=5^{-2}}.$ e) ${2^{(-2)\cdot(-3)}=2^6}.$ f) ${3^{-4-2}=3^{-6}}.$

3. Calculate: a) $7^0;$ b) $4^{-2};$ c) $\left(\frac{3}{5}\right)^{-1};$ d) $\left(-\frac{1}{2}\right)^{-3};$ e) $10^{-4}.$

Answer

a) ${7^0=1}.$ b) ${4^{-2}=\frac{1}{16}}.$ c) ${\left(\frac{3}{5}\right)^{-1}=\frac{5}{3}}.$ d) ${\left(-\frac{1}{2}\right)^{-3}=(-2)^3=-8}.$ e) ${10^{-4}=0.0001}.$

4. Calculate: a) $25^{\frac{1}{2}};$ b) $64^{\frac{1}{3}};$ c) $81^{\frac{3}{4}};$ d) $8^{-\frac{2}{3}};$ e) $\left(\frac{4}{9}\right)^{\frac{3}{2}}.$

Answer

a) ${\sqrt{25}=5}.$ b) ${\sqrt[3]{64}=4}.$ c) ${\left(\sqrt[4]{81}\right)^3=3^3=27}.$ d) ${\frac{1}{\left(\sqrt[3]{8}\right)^2}=\frac{1}{4}}.$ e) ${\left(\sqrt{\frac{4}{9}}\right)^3=\left(\frac{2}{3}\right)^3=\frac{8}{27}}.$

5. Write as a power with a fractional exponent (${x\gt0}$): a) $\sqrt{x};$ b) $\sqrt[3]{x^2};$ c) $\frac{1}{\sqrt{x}};$ d) ${\sqrt{x}\cdot\sqrt[4]{x}}.$

Answer

a) $x^{\frac{1}{2}}.$ b) $x^{\frac{2}{3}}.$ c) ${\frac{1}{x^{\frac{1}{2}}}=x^{-\frac{1}{2}}}.$ d) ${x^{\frac{1}{2}}\cdot x^{\frac{1}{4}}=x^{\frac{1}{2}+\frac{1}{4}}=x^{\frac{3}{4}}}.$

6. Simplify (the letters are not zero): a) $(3a^2b)^3;$ b) $\frac{12x^7y^3}{4x^2y^5};$ c) ${(-2x^3)^2\cdot5x};$ d) $\left(\frac{a^3}{2b}\right)^4.$

Answer

a) $27a^6b^3.$ b) ${3x^5y^{-2}=\frac{3x^5}{y^2}}.$ c) ${4x^6\cdot5x=20x^7}.$ d) $\frac{a^{12}}{16b^4}.$

7. Write without negative exponents (the letters are not zero): a) $x^{-4};$ b) $5a^{-3};$ c) $\frac{2}{y^{-2}};$ d) $(2x)^{-3};$ e) $\frac{a^{-2}}{b^{-5}}.$

Answer

a) $\frac{1}{x^4}.$ b) $\frac{5}{a^3}.$ c) $2y^2.$ d) ${\frac{1}{(2x)^3}=\frac{1}{8x^3}}.$ e) $\frac{b^5}{a^2}.$

8. Calculate without a calculator: a) $\frac{2^{10}\cdot2^{-3}}{2^5};$ b) $\frac{9^4\cdot3^5}{27^4};$ c) ${5^{12}\cdot0.2^{12}};$ d) $\frac{6^8}{2^8\cdot3^7}.$

Answer

a) ${2^{10-3-5}=2^2=4}.$

b) ${9^4=(3^2)^4=3^8}$ and ${27^4=(3^3)^4=3^{12}},$ so ${\frac{3^8\cdot3^5}{3^{12}}}={3^{8+5-12}}={3^1}={3.}$

c) ${(5\cdot0.2)^{12}=1^{12}=1}.$

d) ${\frac{2^8\cdot3^8}{2^8\cdot3^7}=3^1=3}.$

9. Calculate ${\left(\frac{1}{2}\right)^{-3}-3^0+16^{\frac{3}{4}}}.$

Answer

${2^3-1+\left(\sqrt[4]{16}\right)^3}={8-1+8}={15.}$

10. Which number is larger? a) $\left(\frac{3}{4}\right)^5$ or $\left(\frac{3}{4}\right)^7;$ b) $5^{-3}$ or $4^{-3};$ c) $8^6$ or $16^4;$ d) $2^{60}$ or $5^{25}.$

Answer

a) The base is less than $1,$ so the smaller exponent gives the larger power: ${\left(\frac{3}{4}\right)^5\gt\left(\frac{3}{4}\right)^7}.$

b) ${5^{-3}=\frac{1}{125}}$ and ${4^{-3}=\frac{1}{64}},$ so ${4^{-3}\gt5^{-3}}.$

c) ${8^6=(2^3)^6=2^{18}}$ and ${16^4=(2^4)^4=2^{16}},$ so ${8^6\gt16^4}.$

d) ${2^{60}=(2^{12})^5=4096^5}$ and ${5^{25}=(5^5)^5=3125^5},$ so ${2^{60}\gt5^{25}}.$

11. Solve the equations: a) ${3^x=81};$ b) ${2^{x+3}=\frac{1}{4}};$ c) ${25^x=125};$ d) ${10^{2x-1}=1000}.$

Answer

a) ${3^x=3^4},$ so ${x=4}.$

b) ${2^{x+3}=2^{-2}},$ so ${x+3=-2}$ and ${x=-5}.$

c) ${5^{2x}=5^3},$ so ${2x=3}$ and ${x=\frac{3}{2}}.$

d) ${10^{2x-1}=10^3},$ so ${2x-1=3}$ and ${x=2}.$