Question

which expression is equivalent to $2^{5} \cdot 3^{5}$?\
$6^{25}$\
$\frac{1}{6^{-5}}$\
$\frac{1}{6^{5}}$\
$6^{10}$

Explanation:

Step1: Recall exponent rule

Use the product of powers rule for exponents: \(a^n \cdot b^n=(a\cdot b)^n\). Here, \(a = 2\), \(b = 3\), \(n = 5\), so \(2^5\cdot3^5=(2\times3)^5=6^5\).

Step2: Simplify each option

• Option 1: \(6^{25}

eq6^5\), incorrect.

• Option 2: Simplify \(\frac{1}{6^{-5}}\). Using the rule \(a^{-n}=\frac{1}{a^n}\), so \(\frac{1}{6^{-5}} = 6^5\), correct.
• Option 3: \(\frac{1}{6^5}=6^{-5}

eq6^5\), incorrect.

• Option 4: \(6^{10}

eq6^5\), incorrect.

Answer:

\(\frac{1}{6^{-5}}\) (the second option)