Simplify using properties of exponents. [5x^(2/3)][4x^(1/4)]

Verified step by step guidance

1. Step 1: Identify the expression to simplify: \([5x^{2/3}][4x^{1/4}]\).
2. Step 2: Use the property of exponents that states \(a^m \cdot a^n = a^{m+n}\) to combine the exponents of \(x\).
3. Step 3: Add the exponents: \(\frac{2}{3} + \frac{1}{4}\).
4. Step 4: Find a common denominator to add the fractions: \(\frac{2}{3} = \frac{8}{12}\) and \(\frac{1}{4} = \frac{3}{12}\).
5. Step 5: Combine the coefficients and the simplified exponent: \(5 \cdot 4 = 20\) and \(x^{\frac{8}{12} + \frac{3}{12}} = x^{\frac{11}{12}}\).

Key Concepts

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the power of a product ( (ab)^n = a^n * b^n). Understanding these properties is essential for simplifying expressions with exponents.

Multiplying Monomials

Multiplying monomials involves combining coefficients and adding the exponents of like bases. For example, when multiplying 5x^(2/3) and 4x^(1/4), you multiply the coefficients (5 and 4) and apply the properties of exponents to the x terms. This process simplifies the expression into a single monomial.

Finding a Common Denominator

When dealing with exponents that have different fractional powers, finding a common denominator can facilitate simplification. This involves rewriting the exponents with a common base, allowing for easier addition or comparison. For instance, converting 2/3 and 1/4 to a common denominator helps in combining the exponents accurately.