Calculate each value mentally. 15⁴/5⁴

Verified step by step guidance

1. Recognize that the expression is a fraction with exponents: \(\frac{15^4}{5^4}\).
2. Recall the property of exponents that states \(\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n\) when the exponents are the same.
3. Apply this property to rewrite the expression as \(\left(\frac{15}{5}\right)^4\).
4. Simplify the base inside the parentheses: \(\frac{15}{5} = 3\), so the expression becomes \$3^4$.
5. Calculate \$3^4\( by multiplying 3 by itself four times: \)3 \times 3 \times 3 \times 3$ (you can do this mentally).

Key Concepts

Properties of Exponents

This concept involves rules for manipulating expressions with exponents, such as the quotient rule which states that when dividing powers with the same exponent, you can divide the bases and keep the exponent. For example, (a^n)/(b^n) = (a/b)^n.

Simplifying Expressions

Simplifying involves reducing expressions to their simplest form by applying arithmetic operations and exponent rules. In this problem, recognizing that both numerator and denominator have the same exponent allows simplification before calculation.

Mental Math Strategies

Mental math techniques help perform calculations quickly without paper. Recognizing patterns, such as factoring common terms or using exponent properties, enables efficient computation of expressions like 15^4/5^4.