Problem 26 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ 4^{3 / 2} $$

Step-by-Step Solution

Answer

The value of the expression \(4^{3 / 2}\) is 8

Step 1: Rewrite The Expression In Radical Form

Rewrite the expression \(4^{3 / 2}\) in radical form. In general, \(a^{m / n} = \sqrt[n]{a^{m}}\). So, the given expression can be rewritten as \(\sqrt[2]{4^{3}}\).

Step 2: Evaluate The Inside Of The Radical

Evaluate the inside of the radical first, which is \(4^{3} = 64\)

Step 3: Determine The Square Root Of The Result

Now that we have the value inside the radical, we can compute the square root of 64, which is 8

Key Concepts

Radical Expressions

Radical expressions are powerful tools in mathematics used to simplify complex expressions. A radical expression involves roots, typically a square root or cube root, and is represented by the radical symbol \(\sqrt{}\). This makes the expression represent the root of a number or another expression.

When dealing with rational exponents, you can rewrite them as radical expressions. For example, if you have an expression like \(a^{m/n}\), it translates to taking the \(n^{th}\) root of \(a^m\), represented as \(\sqrt[n]{a^m}\). This concept is crucial because it allows us to simplify and evaluate expressions with roots without using complex calculations.

• Radicals help in expressing roots in a simplified form.
• They are used to convert rational exponents into easier formats.

Understanding radical expressions can make working with powers and roots much more intuitive.

Exponents

Exponents are a fundamental concept in mathematics, expressing how many times a number, known as the base, is multiplied by itself. For example, in \(b^n\), \(b\) is the base, and \(n\) is the exponent.

Exponents can be whole numbers, fractions, or even negative numbers. Rational exponents, like \(3/2\), show both an exponential power and a root. For instance, \(4^{3/2}\) means you multiply 4 by itself 3 times, then take the square root of the result.

• Rational exponents combine powers and roots, offering more flexibility.
• Expressions like \(4^{3/2}\) simplify calculations with large powers.

Mastering exponents leads to a deep understanding of polynomial expressions and the ability to solve complex equations.

Square Root

The square root of a number is a special type of root where you are essentially reversing the process of squaring a number. If you have a number \(a\), its square root, denoted as \(\sqrt{a}\), is a value that, when multiplied by itself, gives \(a\).

For example, the square root of 64 is 8 because \(8 \times 8 = 64\). Square roots are particularly important when dealing with expressions that involve radical and rational exponents, like \(4^{3/2}\).

• Finding the square root is essential for simplifying expressions.
• Understanding square roots helps in evaluating expressions like \(\sqrt[2]{4^3}\).

Knowing how to calculate and interpret square roots is a cornerstone of solving problems involving powers and roots.