Determine whether each statement is true or false. If false, correct the right side of the equation. (m^2/3)(m^1/3) = m^2/9

Verified step by step guidance

1. Recall the property of exponents that states when multiplying expressions with the same base, you add the exponents: \(a^{x} \times a^{y} = a^{x+y}\).
2. Identify the base and exponents in the given expression: the base is \(m\), and the exponents are \(\frac{2}{3}\) and \(\frac{1}{3}\).
3. Add the exponents: \(\frac{2}{3} + \frac{1}{3} = \frac{2+1}{3} = \frac{3}{3}\).
4. Simplify the sum of the exponents: \(\frac{3}{3} = 1\).
5. Rewrite the product using the sum of exponents: \((m^{\frac{2}{3}})(m^{\frac{1}{3}}) = m^{1} = m\). Therefore, the original statement is false, and the correct right side of the equation is \(m\).

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers of the same base. Specifically, when multiplying like bases, you add the exponents: a^m * a^n = a^(m+n). This rule is essential for correctly combining terms like m^(2/3) and m^(1/3).

Fractional Exponents

Fractional exponents represent roots and powers simultaneously. For example, m^(1/3) means the cube root of m, and m^(2/3) means the square of the cube root of m. Understanding how to add fractional exponents requires knowledge of adding fractions with common denominators.

Simplifying Exponent Expressions

Simplifying exponent expressions involves correctly performing arithmetic on the exponents after applying the laws of exponents. In this problem, adding 2/3 and 1/3 yields 3/3 or 1, so the product simplifies to m^1 = m, not m^(2/9). Recognizing and correcting such errors is key.