Calculate each value mentally. (20^3/2)/(5^3/2)

Verified step by step guidance

1. Rewrite the expression using fractional exponents: \(\frac{20^{\frac{2}{3}}}{5^{\frac{3}{2}}}\).
2. Express the bases in terms of prime factors or simpler components if possible. For example, \(20 = 2^2 \times 5\) and \(5\) is already prime.
3. Apply the exponent to each factor in the numerator: \(20^{\frac{2}{3}} = (2^2 \times 5)^{\frac{2}{3}} = 2^{\frac{4}{3}} \times 5^{\frac{2}{3}}\).
4. Rewrite the denominator with the exponent applied: \(5^{\frac{3}{2}}\) remains as is.
5. Combine the expression as \(\frac{2^{\frac{4}{3}} \times 5^{\frac{2}{3}}}{5^{\frac{3}{2}}}\) and simplify the powers of 5 by subtracting the exponents: \(5^{\frac{2}{3} - \frac{3}{2}}\).

Key Concepts

Rational Exponents

Rational exponents represent roots and powers combined, where the numerator is the power and the denominator is the root. For example, a^(m/n) means the nth root of a raised to the mth power, or (√[n]{a})^m. Understanding this helps simplify expressions involving fractional powers.

Properties of Exponents

Properties of exponents include rules like a^m * a^n = a^(m+n), (a^m)^n = a^(m*n), and a^m / a^n = a^(m-n). These rules allow us to manipulate and simplify expressions with exponents, especially when dividing or multiplying terms with the same base.

Simplifying Expressions with Different Bases

When simplifying expressions with different bases, it can help to rewrite bases as powers of a common base if possible. For example, 20 and 5 can be expressed in terms of prime factors to apply exponent rules effectively, enabling easier mental calculation.