Question

complete the table
rewrite each expression as a single power.
expression | single power
$\frac{7^{5} cdot 7^{2}}{7^{3}}$ | $7^{4}$
$\frac{4^{9}}{4^{2} cdot 4^{4}}$ |
$\frac{2^{3} cdot 2^{3} cdot 2^{3}}{2 cdot 2 cdot 2}$ |
$2^{4} cdot 3^{4}$ |
$\frac{6^{7}}{2^{7}}$ |
$\frac{(8^{4})^{2}}{8}$ |

Explanation:

Step1: Simplify denominator first

For $\frac{4^9}{4^2 \cdot 4^4}$, use product rule $a^m \cdot a^n = a^{m+n}$:
$4^2 \cdot 4^4 = 4^{2+4}=4^6$

Step2: Apply quotient rule $\frac{a^m}{a^n}=a^{m-n}$

$\frac{4^9}{4^6}=4^{9-6}=4^3$

Step3: Simplify numerator first

For $\frac{2^3 \cdot 2^3 \cdot 2^3}{2 \cdot 2 \cdot 2}$, numerator: $2^3 \cdot 2^3 \cdot 2^3=2^{3+3+3}=2^9$
Denominator: $2 \cdot 2 \cdot 2=2^{1+1+1}=2^3$

Step4: Apply quotient rule

$\frac{2^9}{2^3}=2^{9-3}=2^6$

Step5: Check base compatibility

For $2^4 \cdot 3^4$, use power of product rule $a^n \cdot b^n=(a \cdot b)^n$:
$2^4 \cdot 3^4=(2 \cdot 3)^4=6^4$

Step6: Apply quotient of powers rule

For $\frac{6^7}{2^7}$, use $\frac{a^n}{b^n}=(\frac{a}{b})^n$:
$\frac{6^7}{2^7}=(\frac{6}{2})^7=3^7$

Step7: Simplify numerator first

For $\frac{(8^4)^2}{8}$, use power of power rule $(a^m)^n=a^{m \cdot n}$:
$(8^4)^2=8^{4 \times 2}=8^8$

Step8: Apply quotient rule

$\frac{8^8}{8}=8^{8-1}=8^7$

Answer:

Expression	Single Power
$\frac{4^9}{4^2 \cdot 4^4}$	$4^3$
$\frac{2^3 \cdot 2^3 \cdot 2^3}{2 \cdot 2 \cdot 2}$	$2^6$
$2^4 \cdot 3^4$	$6^4$
$\frac{6^7}{2^7}$	$3^7$
$\frac{(8^4)^2}{8}$	$8^7$