Question

Simplify each expression. $$100^{3 / 2}$$

Short answer

1000

Step-by-step solution

Understand the Problem

The goal is to simplify the expression 100^{3 / 2}. We need to break this down into more manageable parts and simplifly each part.

Rewrite the Expression

Rewrite 100^{3 / 2} using the property of exponents, which states that a fractional exponent can be interpreted as a root and a power. Thus, 100^{3 / 2} is the same as \( \sqrt{100}^3 \).

Calculate the Square Root

Find the square root of 100. \( \sqrt{100} = 10\).

Raise to the Third Power

Raise 10 to the third power. \( 10^3 = 1000 \).

Write the Simplified Form

Combine the results to write the final simplified form of the original expression, \(100^{3 / 2} = 1000 \).

Key concepts

Fractional Exponents

Fractional exponents can seem tricky at first, but they are really just another way to express roots and powers. When you see an exponent in the form of a fraction, like \(a^{m/n}\), it means you're dealing with both a root and a power. The denominator ( \(n\) ) indicates the type of root you're taking (\(n\)-th root), while the numerator ( \(m\) ) shows the power. So, \(100^{3/2}\) can be reworked as \( (\sqrt{100})^3 \). Understanding this helps you break down problems into more manageable parts. It's like rephrasing the problem in a language you understand better.

Here are a few key points about fractional exponents:

• Fractional exponents combine roots and powers
• The denominator indicates the root, while the numerator indicates the power
• This helps in simplifying complex expressions step-by-step

Square Roots

Square roots are a special case of roots where you're looking for a number that, when multiplied by itself, gives you the original number. For example, the square root of 100 is 10, because \( 10 \times 10 = 100 \). In the given problem, we use the square root as part of simplifying the expression \(100^{3/2}\). When rephrased, this becomes \((\sqrt{100})^3\), which allows us to take the easier route of finding the square root first and then raising it to the power.

Important points to remember about square roots:

• The square root of \(a\) is a number \(b\) such that \(b^2 = a\)
• It is denoted as \(\sqrt{a}\)
• Square roots are widely used in solving quadratic equations and other algebraic problems

Properties of Exponents

Exponents follow certain properties that make operations easier. Knowing these properties helps in breaking down and simplifying complex expressions. Here are some crucial properties that are useful:

• \(a^{m/n} = (\sqrt[n]{a})^m\)
• \((a^m)^n = a^{mn}\)
• \(a^m \times a^n = a^{m+n}\)
• \(a^0 = 1\)

In the problem we are solving, the property \(a^{m/n} = (\sqrt[n]{a})^m\) is directly applied to simplify \(100^{3/2}\) into \((\sqrt{100})^3\). By using these exponent properties, what seems like a complicated problem can be broken down into simple, easy solvable steps.

Powers of Numbers

Raising numbers to a power means multiplying that number by itself a certain number of times, indicated by the power. For example, \(2^3\) means \(2 \times 2 \times 2 = 8\). Powers are a way to express repeated multiplication compactly. In the exercise \(100^{3/2}\), we already found that \(\sqrt{100} = 10\). Raising 10 to the third power \(10^3\) involves multiplying 10 by itself three times: \(10 \times 10 \times 10 = 1000\).

Key notes on powers of numbers:

• A power is repeated multiplication of a number
• A number raised to the power of 1 is the number itself
• Zero to any power is zero, and any number to the power of zero is one
• Powers make it easy to express very large or very small numbers