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 \(a^m \cdot a^n = a^{m+n}\).
2. Identify the base and exponents in the given expression: the base is \(y\), and the exponents are 2 and 5 in \((y^2)(y^5)\).
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. Since the right side of the equation is \(y^7\), the statement is true.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers of the same base. Specifically, when multiplying powers with the same base, you add their exponents. For example, y^a * y^b = y^(a+b). This rule is essential for simplifying expressions like (y^2)(y^5).

Base Consistency in Exponentiation

Exponent rules apply only when the bases are the same. In the expression (y^2)(y^5), both terms have the base y, so the exponents can be combined. If the bases differ, the exponents cannot be directly added or subtracted.

Evaluating True or False Statements in Algebra

Determining whether an algebraic statement is true or false involves applying relevant rules and verifying the equality. If false, the correct form must be found by properly applying algebraic principles, such as exponent laws, to rewrite the expression accurately.