Simplify using properties of exponents. $\left(5x^{\frac{2}{3}}\right)\left(4x^{\frac{1}{4}}\right)$

Verified step by step guidance

1. Identify the expression to simplify: \([5x^{2\/3}][4x^{1\/4}]\).
2. Use the property of exponents that states when multiplying terms with the same base, you add the exponents: \(x^{a} \cdot x^{b} = x^{a+b}\).
3. Multiply the coefficients (numbers) separately: \(5 \times 4\).
4. Add the exponents of \(x\): \(\frac{2}{3} + \frac{1}{4}\).
5. Write the simplified expression as the product of the multiplied coefficients and the base with the new exponent: \((5 \times 4) x^{(\frac{2}{3} + \frac{1}{4})}\).

Key Concepts

Properties of Exponents

Properties of exponents are rules that simplify expressions involving powers. Key rules include multiplying powers with the same base by adding their exponents, and raising a power to another power by multiplying exponents. These properties help combine and simplify expressions efficiently.

Multiplying Expressions with Exponents

When multiplying expressions with the same base, you add the exponents while keeping the base unchanged. For example, x^a * x^b = x^(a+b). This rule is essential for simplifying products of exponential terms.

Fractional Exponents

Fractional exponents represent roots and powers simultaneously. For instance, x^(m/n) means the nth root of x raised to the mth power. Understanding fractional exponents allows you to manipulate and simplify expressions involving roots and powers.