Write each root using exponents and evaluate. ∛125

Verified step by step guidance

1. Recognize that the cube root of a number can be expressed as an exponent: \( \sqrt[3]{125} = 125^{1/3} \).
2. Identify that 125 is a power of 5, specifically \( 125 = 5^3 \).
3. Substitute \( 5^3 \) for 125 in the expression: \( (5^3)^{1/3} \).
4. Apply the power of a power property of exponents, which states \( (a^m)^n = a^{m \cdot n} \).
5. Calculate the exponent: \( 5^{3 \cdot (1/3)} = 5^1 \).

Key Concepts

Roots and Exponents

Roots and exponents are fundamental concepts in algebra that describe the relationship between numbers. The nth root of a number is a value that, when raised to the nth power, gives the original number. For example, the cube root (∛) of 125 is the number that, when multiplied by itself three times, equals 125. This can also be expressed using exponents as 125^(1/3).

Evaluating Roots

Evaluating roots involves finding the numerical value of a root expression. In the case of ∛125, we seek a number that, when cubed, results in 125. This requires understanding perfect cubes, as 5^3 equals 125, thus ∛125 evaluates to 5. This process is essential for simplifying expressions and solving equations involving roots.

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers, quotient of powers, and power of a power. These rules help simplify expressions and solve equations efficiently. For instance, knowing that a^(m/n) can be rewritten as the nth root of a raised to the m power aids in understanding and evaluating expressions like 125^(1/3).