Question

Simplify each expression. $$25^{3 / 2}$$

Short answer

The simplified expression is 125.

Step-by-step solution

- Express the Exponent in Radical Form

The expression has a rational exponent. Rewrite the exponent as a radical to understand it better. We have: \[ 25^{3 / 2} = (25^{1 / 2})^3 \] This expresses the base 25 raised to the power of 3/2 as a square root raised to the third power.

- Simplify the Inner Radical

Simplify the expression inside the radical. We have: \[ 25^{1 / 2} = \sqrt{25} \] And we know that \sqrt{25} = 5.

- Raise the Result to the Power of 3

Now we take the result of \sqrt{25} and raise it to the 3rd power. We have: \[ 5^3 = 5 \times 5 \times 5 = 125 \]

Key concepts

Rational Exponents

Rational exponents are a way to express powers and roots in one term. For example, when you see something like 25^(3/2), it might seem confusing at first. But let's break it down. The numerator (3) is the power, and the denominator (2) is the root. So, 25^(3/2) can be rewritten as the square root of 25 raised to the power of 3.
This is a fundamental concept in algebra as it allows us to switch between exponential forms and radical forms easily, making calculations more flexible. When faced with rational exponents, always remember that the denominator tells you the root, and the numerator tells you the power.

Radicals

Radicals are symbols that indicate roots. The most common radical is the square root, represented by the symbol \, which looks like a checkmark with a line above it. When we see \(\backslash sqrt{n}\), it means we are looking for a number that, when multiplied by itself, gives n.
For instance, \(\backslash sqrt{25}\) asks what number squared equals 25. The answer is 5 because 5*5=25. Radicals are closely related to rational exponents. In fact, rewriting 25^(1/2) as the square root of 25 helps us solve the expression easily.
Radicals can be used with various roots like cube roots, fourth roots, etc., where the root number is indicated inside the radical sign or in the exponent's denominator.

Powers

Powers, or exponents, are a way to express repeated multiplication. When a number (called the base) is raised to an exponent, it is multiplied by itself the number of times indicated by the exponent. For example, 5^3 means 5 * 5 * 5, which equals 125.
Powers are used extensively in algebra and beyond, making calculations simpler and expressions more compact. Understanding how to manipulate powers, like raising a base to a fractional exponent or simplifying powers using radicals, is key to mastering algebra.
By understanding powers and how to express and simplify them, tackling more complex algebraic expressions becomes much easier.