Exponentiation problems: Problems with Solutions

Solution:
We multiply the right side and get [tex]3^2 \times 3^3=3^{2+3}=3^5=3^n[/tex], so [tex]n=5[/tex].

Solution:
We have [tex]2^{-3}=(2^3)^{-1}=8^{-1}=8^n[/tex], so [tex]n=-1[/tex].

Solution:
[tex]3^4=3 \times 3 \times 3 \times 3=9 \times 3 \times 3=27 \times 3=81[/tex]

Solution:
[tex]2^8=2^{2 \times 4}=(2^2)^4=4^4=4 \times 4 \times 4 \times 4=16 \times 4 \times 4=64 \times 4=256[/tex]

Solution:
[tex]2^3 \times 3^2=2 \times 2 \times 2 \times 3 \times 3=4 \times 2 \times 3 \times 3=8 \times 3 \times 3=24 \times 3=72[/tex]

Solution:
[tex]5^2 \times 2^3 \times 3=(5 \times 5) \times (2 \times 2 \times 2) \times 3=25 \times 8 \times 3=200 \times 3=600[/tex]

Solution:
[tex]3^4-3^3=3^3 \times 3-3^3 \times 1=3^3(3-1)=3^3 \times 2=3 \times 3 \times 3 \times 2=(3 \times 3) \times (3 \times 2)=9 \times 6=54[/tex]

Solution:
[tex]5 \times 3^4-2 \times 3^5=5 \times 3^4-2 \times 3 \times 3^4=5 \times 3^4-6 \times 3^4=-1 \times 3^4=-3 \times 3 \times 3 \times 3=-81[/tex]

Solution:
[tex](-2)^{3} = (-2) \times (-2) \times (-2) = -8[/tex]

Solution:
[tex]5^{0} = 1[/tex]

Solution:
[tex]2^{-3}=\frac{1}{2^3}=\frac{1}{8}[/tex]

Solution:
Every nonzero number raised to the power [tex]0[/tex] equals [tex]1[/tex].
[tex]4^2=16[/tex], [tex]4^0=1[/tex], [tex](-2)^4=16[/tex]
[tex]16+1-16=1[/tex]

Solution:
Be careful: the powers are calculated first; [tex]-(-2)^3[/tex] means the opposite of [tex](-2)^3[/tex].
[tex](-2)^3=-8[/tex], [tex](-2)^2=4[/tex]
[tex]-(-8)-4+5=8-4+5=9[/tex]

Solution:
[tex]x\times x\times x=x^3[/tex]
[tex]27\,000=27\times 1000=3^3\times 10^3=30^3[/tex]
[tex]x^3=30^3 \Rightarrow x=30[/tex]

Solution:
[tex]6^5=(2\times 3)^5=2^5\times 3^5[/tex]
[tex]2^5\times 3^5\div 3^4=2^5\times 3=32\times 3=96[/tex]

Solution:
[tex]2^{15}+2^{15}+2^{15}+2^{15}=4\times 2^{15}=2^2\times 2^{15}=2^{17}[/tex]
[tex]2^7\times 2^8=2^{15}[/tex]
[tex]A=\frac{2^{17}}{2^{15}}=2^2=4[/tex]

Solution:
Take the common factor [tex]3^6[/tex] out of the numerator and the denominator:
[tex]\frac{3^6(1+3^2)}{3^6(1+3)}=\frac{10}{4}=\frac{5}{2}=2\frac{1}{2}[/tex]

Solution:
[tex]27x^3=3^3x^3=(3x)^3[/tex]
[tex]y^6=(y^2)^3[/tex]
[tex]\frac{27x^3}{y^6}=\frac{(3x)^3}{(y^2)^3}=\left(\frac{3x}{y^2}\right)^3[/tex]

Solution:
[tex]4^{20}=(2^2)^{20}=2^{40}[/tex]
[tex]\frac{2^{40}}{2}=2^{39}[/tex]
Be careful: half of [tex]4^{20}[/tex] is not [tex]2^{20}[/tex] – halving the base does not halve the number.