Simplify: (5⁻² × 25^(x−1)) / 125^x

Question

ScanSolve AI Tutor · Accepted Answer

Step 1: Rewrite all bases as powers of 5. We know that 25 = 5² and 125 = 5³. Substitute these into the expression. (5⁻² × (5²)^(x−1))/((5³)^(x+1)) Step 2: Apply the power rule (a^m)ⁿ = a^(mn) to simplify the exponents. (5⁻² × 5^(2(x−1)))/(5^(3(x+1))) = (5⁻² × 5^(2x−2))/(5^(3x+3)); Step 3: Apply the multiplication rule a^m × aⁿ = a^(m+n) to the numerator. (5^(−2 + (2x−2)))/(5^(3x+3)) = (5^(2x−4))/(5^(3x+3)); Step 4: Apply the division rule (a^m)/(aⁿ) = a^(m−n) to the entire expression. 5^((2x−4) − (3x+3)) Step 5: Simplify the exponent by distributing the negative sign and combining like terms. 5^(2x−4−3x−3) = 5^((2x−3x) + (−4−3)) = 5^(−x−7) The simplified expression is: 5^(−x−7)

Simplify: (5⁻² * 25^(x-1)) / 125^x

Answer

Step-by-step solution

Rewrite all bases as powers of 5.

Apply the power rule $(a^m)^n = a^{mn}$ to simplify the exponents.

Apply the multiplication rule $a^m \times a^n = a^{m+n}$ to the numerator.

Apply the division rule $\frac{a^m}{a^n} = a^{m-n}$ to the entire expression.

Simplify the exponent by distributing the negative sign and combining like terms.

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