Calculate each value mentally. 15⁴/5⁴

Verified step by step guidance

1. Recognize that the expression is a division of two powers with the same exponent: \(\frac{15^{4}}{5^{4}}\).
2. Use the property of exponents that states \(\frac{a^{n}}{b^{n}} = \left(\frac{a}{b}\right)^{n}\) to rewrite the expression as \(\left(\frac{15}{5}\right)^{4}\).
3. Simplify the fraction inside the parentheses: \(\frac{15}{5} = 3\), so the expression becomes \$3^{4}$.
4. Recall that \$3^{4}$ means multiplying 3 by itself 4 times: \(3 \times 3 \times 3 \times 3\).
5. Calculate the value of \$3^{4}$ mentally by performing the multiplications step-by-step.

Key Concepts

Properties of Exponents

This concept involves rules for manipulating expressions with exponents, such as the quotient rule which states that when dividing like bases, you subtract the exponents. Understanding these properties helps simplify expressions like (15^4)/(5^4) by recognizing common bases or rewriting terms.

Simplifying Exponential Expressions

Simplifying exponential expressions means reducing them to their simplest form by applying exponent rules and arithmetic operations. For example, rewriting (15^4)/(5^4) as (15/5)^4 allows easier mental calculation by simplifying the base before applying the exponent.

Mental Math Strategies

Mental math strategies involve techniques to perform calculations quickly without paper, such as breaking down numbers into factors or using exponent rules to simplify expressions. Applying these strategies to (15^4)/(5^4) enables efficient computation by simplifying the base first.