Determine whether each statement is true or false. If false, correct the right side of the equation. (y²)(y⁵) = y⁷

Verified step by step guidance

1. Recall the product rule for exponents: when multiplying expressions with the same base, you add the exponents. Mathematically, this is written as \(a^{m} \times a^{n} = a^{m+n}\).
2. Identify the base and exponents in the given expression: \((y^{2})(y^{5})\). Here, the base is \(y\), and the exponents are 2 and 5.
3. Apply the product rule by adding the exponents: \$2 + 5 = 7$.
4. Rewrite the expression using the sum of the exponents: \((y^{2})(y^{5}) = y^{7}\).
5. Conclude that the original statement is true because the right side correctly represents the product of the left side using the product rule for exponents.

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 their exponents: a^m * a^n = a^(m+n). This rule is essential for correctly simplifying expressions like (y^2)(y^5).

Base Consistency in Exponentiation

For exponent rules to apply directly, the bases must be identical. In the expression (y^2)(y^5), both terms have the base y, allowing the exponents to be combined. Recognizing the base ensures proper application of exponent laws.

Evaluating True or False Statements in Algebra

Determining the truth of algebraic statements involves verifying if both sides of an equation are equivalent under given rules. If false, the correct form must be identified by applying relevant algebraic principles, such as exponent laws in this case.