Question

rewriting powers
zakia james
card sort
sort the expressions based on whether they are equivalent to $6^8$.
equivalent to $6^8$
not equivalent to $6^8$
$\frac{6^4 \cdot 6^4}{6^1}$
$\frac{12^8}{2^8}$
$2^3 \cdot 3^5$
$\frac{6^7 \cdot 6^7}{(6^3)^2}$
$2^8 \cdot 3^8$
5 of 10

Explanation:

Step1: Simplify $2^8 \cdot 3^8$

Use exponent rule $a^n \cdot b^n=(a \cdot b)^n$:
$2^8 \cdot 3^8=(2 \cdot 3)^8=6^8$

Step2: Simplify $\frac{6^4 \cdot 6^4}{6^1}$

First, add exponents in numerator: $6^4 \cdot 6^4=6^{4+4}=6^8$.
Then subtract exponents for division: $\frac{6^8}{6^1}=6^{8-1}=6^7$

Step3: Simplify $\frac{12^8}{2^8}$

Rewrite $12=2 \cdot 6$, so $12^8=(2 \cdot 6)^8=2^8 \cdot 6^8$.
Divide: $\frac{2^8 \cdot 6^8}{2^8}=6^8$

Step4: Simplify $2^3 \cdot 3^5$

Calculate values: $2^3=8$, $3^5=243$.
Product: $8 \cdot 243=1944$.
$6^8=1679616$, so $1944
eq 1679616$

Step5: Simplify $\frac{6^7 \cdot 6^7}{(6^3)^2}$

Add exponents in numerator: $6^7 \cdot 6^7=6^{7+7}=6^{14}$.
Simplify denominator: $(6^3)^2=6^{3 \cdot 2}=6^6$.
Divide: $\frac{6^{14}}{6^6}=6^{14-6}=6^8$

Answer:

Equivalent to $6^8$:

$2^8 \cdot 3^8$, $\frac{12^8}{2^8}$, $\frac{6^7 \cdot 6^7}{(6^3)^2}$

Not Equivalent to $6^8$:

$\frac{6^4 \cdot 6^4}{6^1}$, $2^3 \cdot 3^5$