MENU
❌
Home
Math Forum/Help
Problem Solver
Practice
Worksheets
Tests
Algebra
Geometry
College Math
History
Games
MAIN MENU
1 Grade
Adding and subtracting up to 10
Comparing numbers up to 10
Adding and subtracting up to 20
Addition and Subtraction within 20
2 Grade
Adding and Subtracting up to 100
Addition and Subtraction within 20
3 Grade
Addition and Subtraction within 1000
Multiplication up to 5
Multiplication Table
Rounding
Dividing
Perimeter
Addition, Multiplication, Division
4 Grade
Adding and Subtracting
Addition, Multiplication, Division
Equivalent Fractions
Divisibility by 2, 3, 4, 5, 9
Area of Squares and Rectangles
Fractions
Equivalent Fractions
Least Common Multiple
Adding and Subtracting
Fraction Multiplication and Division
Operations
Mixed Numbers
Decimals
Expressions
6 Grade
Percents
Signed Numbers
The Coordinate Plane
Equations
Expressions
Polynomials
Polynomial Vocabulary
Symplifying Expressions
Polynomial Expressions
Factoring
7 Grade
Angles
Inequalities
Linear Functions
8 Grade
Congurence of Triangles
Linear Functions
Systems of equations
Slope
Parametric Linear Equations
Word Problems
Exponents
Roots
Quadratic Equations
Quadratic Inequalities
Rational Inequalities
Vieta's Formulas
Progressions
Arithmetic Progressions
Geometric Progression
Progressions
Number Sequences
Reciprocal Equations
Logarithms
Logarithmic Expressions
Logarithmic Equations
Logarithmic Equations
Logarithmic Inequalities
Irrational Equations
Irrational Inequalities
Trigonometry
Trigonometry
Identities
Trigonometry
Trigonometric Equations
Trigonometric Inequalities
Extremal value problems
Numbers Classification
Geometry
Slope
Intercept Theorem
Law of Sines
Law of Cosines
Vectors
Modulus Inequalities
Exponential Inequalities
Exponential Equations
Modulus equations
Probabilities
Functions
Min, Max Values
Limits
Limits of Functions
Monotonicity of Functions
Properties of Triangles
Pythagorean Theorem
Matrices
Complex Numbers
Inverse Trigonometric Functions
Analytic Geometry
Analytic Geometry
Circle
Parabola
Ellipse
Conic sections
Polar coordinates
Derivatives
Derivatives
Applications of Derivatives
Derivatives
Integrals
Integrals
Integration by Parts
Trigonometric Substitutions
Application
Differential Equations
Home
Practice
Exponents and Radicals
Easy
Normal
Exponents and Radicals: Problems with Solutions
Problem 1
Which of the following is true?
$7^{2}<3^{4}$
$7^{2}>3^{4}$
$\ 7^{2}=3^{4}$
$7^{2}\geq 3^{4}$
Solution:
$7^{2}<3^{4}$ because $49<81$
Problem 2
Which of the following is true?
$4^{2}<2^{4}$
$4^{2}>2^{4}$
$4^{2}=2^{4}$
$4^{2}\geq 2^{4}$
Solution:
$4^{2}=2^{4}$ because $16=16$
Problem 3
Which of the following is true?
$3^{5}<5^{3}$
$3^{5}>5^{3}$
$3^{5}=5^{3}$
$3^{5}\geq 5^{3}$
Solution:
$3^{5}>5^{3}$ because $243>125$
Problem 4
Which property is used?
$3^{5} \cdot 3^{2}\cdot 3^{-3}=3^{4}$
Solution:
Property 1 is used.
$x^{n} \cdot x^{m}=x^{n+m}\Longrightarrow 3^{5}\cdot 3^{2} \cdot 3^{-3}=3^{5+2-3}=3^{4}$
Problem 5
Which property is used?
$\frac{2^{12}}{2^{8}}=16$
Solution:
Property 2 is used.
$\frac{x^{n}}{x^{m}}=x^{n-m}$
$\frac{2^{12}}{2^{8}}=2^{12-8}=2^{4}=16$
Problem 6
Which property is used?
$(2^{2})^{3}=64$
Solution:
Property 3 is used.
$(x^{n})^{m}=x^{n\cdot m}$
$(2^{2})^{3}=2^{6}=64$
Problem 7
Which property is used?
$\left( \frac{5}{2}\right) ^{3}=\frac{125}{8}$
Solution:
Property 5 is used.
$(\frac{x}{y})^{n}=\frac{x^{n}}{y^{n}}$
$\left( \frac{5}{2}\right) ^{3}=\frac{5^{3}}{2^{3}}=\frac{125}{8}$
Problem 8
$5^{2}\cdot 5^{3}=$
$5^6$
$5^5$
$5^4$
$6^5$
Solution:
If we use property 1, we have
$5^{2} \cdot 5^{3}=5^{2+3}=5^{5}$.
Problem 9
Which property is used?
$(2\times 3)^{3}=216$
Solution:
Property 4 is used.
$(x \cdot y)^{n}=x^{n}\cdot y^{n}$ $(2\times 3)^{3}=2^{3}\times 3^{3}=8\times 27=216$
Problem 10
Find the value of x if
$3^{10}=3^{6}\cdot 3^{x}$
Solution:
If we use property 1, we have
$3^{10}=3^{6} \cdot 3^{x}=3^{6+x}$
$3^{10}=3^{6+x}$
$10=6+x$
$x=4$
Problem 11
$a^{0}\cdot a^{8}=$
$a$
$a^{8}$
$a^{0\cdot 8} = a^0$
$a^{4}$
Solution:
$a^{0}\cdot a^{8}=a^{0+8}=a^{8}$
Problem 12
$(a^{5})^{6}=$
$a^{11}$
$a^{30}$
$a^{36}$
$a^{25}$
Solution:
$(a^{5})^{6}=a^{5\cdot 6}=a^{30}$
Problem 13
Find the value of n.
$(3^{2})^{3}=n^{6}$
Solution:
$(3^{2})^{3}=3^{6}=n^{6}\Longrightarrow n=3$
Problem 14
$\frac{4^{8}}{4^{4}}=$
$4^2$
$2$
$4$
$4^4$
Solution:
$\frac{4^{8}}{4^{4}}=4^{8-4}=4^{4}$
Problem 15
$\frac{a\cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a}{a \cdot a}=$
$a^5$
$a^6$
$a^7$
$a^8$
Solution:
If we use property 2, we have
$\frac{a\cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a}{a\cdot a}=\frac{a^{9}}{a^{2}}=a^{7}$
Problem 16
$\frac{5^{6}}{5^{-1}}=$
$5^{5}$
$5^{7}$
$5^{-6}$
$\frac{1}{5^6}$
Solution:
$\frac{5^{6}}{5^{-1}}=5^{6-(-1)}=5^{7}$
Problem 17
$\frac{x^{12}}{x^{6}}=$
$x^{9}$
$x^{2}$
$x^{6}$
$x^{4}$
Solution:
$\frac{x^{12}}{x^{6}}=x^{12-6}=x^{6}$
Problem 18
Find The value of x.
$\frac{9^{x}}{9^{2}}=9^{5}$
Solution:
$\frac{9^{x}}{9^{2}}=9^{x-2}=9^{5}\Longrightarrow x-2=5\Longrightarrow x=7$
Problem 19
$(2x^{3})^{4}=$
$16x^{12}$
$4x^{12}$
$4x^{7}$
$16x^{7}$
Solution:
$2^{4}(x^{3})^{4}=16x^{12}$
Problem 20
$(\frac{z^{2}}{2})^{3}=$
$\frac{z^{5}}{8}$
$\frac{z^{5}}{2}$
$\frac{z^{6}}{2}$
$\frac{z^{6}}{8}$
Solution:
If we use property 5, we have
$(\frac{z^{2}}{2})^{3}=\frac{(z^{2})^{3}}{2^{3}}=\frac{z^{6}}{8}$
Problem 21
$(2\cdot y^{4})^{3}=$
$8y^{4}$
$8y^{7}$
$8y^{12}$
$6y^{12}$
Solution:
$(2\cdot y^{4})^{3}=2^{3}\cdot (y^{4})^{3}=8y^{12}$.
Problem 22
$x^{1+\frac{5}{3}}=$
$\sqrt[3]{x^{6}}$
$\sqrt[3]{x^{8}}$
$x^3\cdot x^{\frac{5}{3}}$
$x+x^{\frac{5}{3}}$
Solution:
$x^{1+\frac{5}{3}}=x^{\frac{3+5}{3}}=x^{\frac{8}{3}}=\sqrt[3]{x^{8}}$
Problem 23
$\left( \sqrt[3]{x}\right)^{2}=$
$x^{\frac{3}{2}}$
$x^2$
$\sqrt{x^3}$
$x^{\frac{2}{3}}$
Solution:
$\left( \sqrt[3]{x}\right) ^{2}=\left( x^{\frac{1}{3}}\right)^{2}=x^{\frac{2}{3}}$
Problem 24
$x^{1-\frac{3}{2}+\frac{5}{4}} = $
$\sqrt[4]{x^{3}}$
$\sqrt[3]{x^{4}}$
1
$\sqrt{x^{4}}$
Solution:
$x^{1-\frac{3}{2}+\frac{5}{4}}=x^{\frac{4-6+5}{4}}=x^{\frac{3}{4}}=\sqrt[4]{x^{3}}$
Problem 25
$\left( \sqrt{\sqrt[3]{x}}\right)^{5}=$
$\sqrt[5]{x^{6}}$
$x^{\frac{5}{2}}$
$x^{\frac{5}{3}}$
$x^{\frac{5}{6}}$
Solution:
$\left( \sqrt{\sqrt[3]{x}}\right)^{5}=\left( \left( x^{\frac{1}{3}}\right) ^{\frac{1}{2}}\right) ^{5}=\left( x^{\frac{1}{6}}\right)^{5}=x^{\frac{5}{6}}$
Problem 26
What is the result of $\left( \frac{x^{2}}{2y}\right)^{3}-\left( \frac{x^{2}}{y^{3}}\right)$ ?
$\frac{x^{5}-8x^{2}}{8y^{3}}$
$\frac{x^{6}-8x^{2}}{2y^{3}}$
$\frac{x^{6}-6x^{2}}{6y^{3}}$
$\frac{x^{6}-8x^{2}}{8y^{3}}$
Solution:
$\left( \frac{x^{2}}{2y}\right) ^{3}-\left( \frac{x^{2}}{y^{3}}\right) =\frac{x^{6}}{2^{3}y^{3}}-\frac{x^{2}}{y^{3}}=\frac{x^{6}-8x^{2}}{8y^{3}}$
Problem 27
What is the result of $\left( \frac{x^{2}}{2y}\div \frac{2y^{2}}{3x}\right)^{2}$ ?
$\frac{9x^{6}}{16y^{6}}$
$\frac{3x^{6}}{4y^{6}}$
$\frac{6x^{6}}{8y^{6}}$
$\frac{9x^{4}}{16y^{4}}$
Solution:
$\left( \frac{x^{2}}{2y}\div \frac{2y^{2}}{3x}\right)^{2}=\left(\frac{x^{2}(3x)}{2y^{2}(2y)}\right)^{2}=\left( \frac{3x^{3}}{4y^{3}}\right)^{2}=\frac{9x^{6}}{16y^{6}}$
Problem 28
What is the result of $\left(\frac{ab}{2c}\right)^{3}\div \left( \frac{a.b}{c^{3}}\right)^{2}$ ?
$\frac{abc^{3}}{8}$
$\frac{(abc)^{3}}{8}$
$\frac{abc^{3}}{8}$
$\frac{8}{abc^{3}}$
Solution:
$\left( \frac{a.b}{2c}\right)^{3}\div \left( \frac{a \cdot b}{c^{3}}\right) ^{2}=\frac{a^{3}\cdot b^{3}}{2^{3}c^{3}}\div \frac{a^{2} \cdot b^{2}}{c^{6}}=\frac{a^{3} \cdot b^{3} \cdot c^{6}}{a^{2}\cdot b^{2} \cdot 2^{3}c^{3}}=\frac{a^{3-2} \cdot b^{3-2} \cdot c^{6-3}}{8}=\frac{abc^{3}}{8}$
Problem 29
Evaluate the expression $5^{x}-3^{x}$ when $x=3$.
Solution:
If $x=3$ then
$5^{x}-3^{x}=5^{3}-3^{3}=125-27=98$
Problem 30
What is the value of n if $3^{n}=3^{2}\cdot 3^{3}$
Solution:
$3^{n}=3^{2}\cdot 3^{3}=3^{2+3}=3^{5}$
Therefore, $3^{n}=3^{5}$
So $n=5$
Problem 31
What is the value of $n$ if $8^{n}=2^{-2} \cdot 4^{3}$ ?
$\frac{4}{3}$
$\frac{3}{4}$
$\frac{-6}{3}$
$-6$
Solution:
$8^{n}=2^{-2} \cdot 4^{3}\Longrightarrow (2^{3})^{n}=2^{-2} \cdot (2^{2})^{3}\Longrightarrow 2^{3n}=2^{6-2}\Longrightarrow 2^{3n}=2^{4}$
$3n=4$, so $n=\frac{4}{3}$
Problem 32
What is the value of $x$ if $2^{2x-1}=4$ ?
Solution:
$2^{2x-1}=4\Longrightarrow 2^{2x-1}=2^{2}\Longrightarrow 2x-1=2\Longrightarrow x=\frac{3}{2}$
Problem 33
What is the value of $n$ if $27^{n}=3^{2}\cdot 9^{3}$
$\frac{3}{8}$
$\frac{8}{3}$
$\frac{8}{2}$
$8$
Solution:
$27^{n}=3^{2} \cdot 9^{3}$
$(3^{3})^{n}=3^{2} \cdot(3^{2})^{3}$
$3^{3n}=3^{8}$
$3n=8$
$n=\frac{8}{3}$
Problem 34
What is the value of $x$ if
$\sqrt[3]{8^{x}}=128$
Solution:
$\sqrt[3]{8^{x}}=128$
$\sqrt[3]{(2^{3})^{x}}=2^{7}$
$2^{\frac{3x}{3}}=2^{7}\Longrightarrow x=7$
Easy
Normal
Submit a problem on this page.
Problem text:
Solution:
Answer:
Your name(if you would like to be published):
E-mail(you will be notified when the problem is published)
Notes
: use [tex][/tex] (as in the forum if you would like to use latex).
Correct:
Wrong:
Unsolved problems:
Feedback
Contact email:
Follow us on
Twitter
Facebook
Author
Copyright © 2005 - 2025.