Problem 4 - Algebra and Trigonometry

Question

Explain what \(4^{3 / 2}\) means, then calculate \(4^{3 / 2}\) in two different ways: $$ \left(4^{1 / 2}\right)=-\quad \text { or } \quad\left(4^{3}\right)= $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Exponent Fraction

The expression \(4^{3/2}\) involves a fractional exponent, which means you take the square root of the base and then raise it to the power of the numerator. Specifically, \(4^{3/2}\) is equivalent to \((4^{1/2})^3\) or \((\sqrt{4})^3\).

Step 2: Method 1 - Calculate Using Square Root First

First, evaluate the square root: \(4^{1/2} = \sqrt{4} = 2\). Then raise this result to the power of 3: \(2^3 = 8\). Thus, using this method, \(4^{3/2} = 8\).

Step 3: Method 2 - Calculate Using Large Power First

First, raise the base to the power of 3: \(4^3 = 64\). Then take the square root of this result: \(\sqrt{64} = 8\). Therefore, using this method, \(4^{3/2} = 8\).

Key Concepts

Square Roots

Square roots can be thought of as the inverse operation of squaring a number. When we say the square root of a number, we are looking for which number, when multiplied by itself, gives the original number. For example, the square root of 4 is 2 because when 2 is multiplied by itself (2 x 2), we get 4. We denote the square root of a number \(x\) as \(\sqrt{x}\).
Consider the example given in the exercise: \(4^{3/2}\). The fractional exponent \(3/2\) suggests a square root operation. Specifically, \(4^{1/2} = \sqrt{4}\). By solving, we find that \(\sqrt{4} = 2\).
Understanding square roots is crucial because they frequently appear in problems involving exponents and fractional exponents. Practicing more examples can solidify this concept.

Exponential Calculation

Exponential calculations involve working with numbers raised to a power. The base of the expression is the number being multiplied, and the exponent indicates how many times the base is used in repeated multiplication. For instance, in \(4^3\), 4 is the base and 3 is the exponent, and it means you multiply 4 by itself 3 times (4 x 4 x 4).
In the context of the exercise, calculating \(4^{3/2}\) involves using exponential calculations in a slightly different way. The solution approaches involve either taking a square root first and then raising to a power (Method 1) or computing a power first and then a square root (Method 2).

• Method 1: First, calculate \(4^{1/2} = \sqrt{4} = 2\). Then raise this result to the power of 3. So, \(2^3 = 8\).
• Method 2: Start by calculating \(4^3 = 64\). Then, take the square root \(\sqrt{64} = 8\).

Both methods yield the same result, demonstrating the flexibility and power of exponential calculations with fractional exponents.

Exponents

Exponents, sometimes called powers or indices, represent the number of times a base number is multiplied by itself. For example, \(3^4\) means multiply 3 by itself 4 times: \(3 \times 3 \times 3 \times 3 = 81\). Exponents make it easier and clearer to express repeated multiplication.
When we have fractional exponents, the concept extends to involve roots. A fractional exponent \(a/b\) can be understood as a base raised to a power \(a\), and then we take the \(b\)-th root of the result. This dual action of multiplication (raising to a power) and rooting (taking a root) is what allows us to express and compute with fractional exponents.
So in \(4^{3/2}\), the base 4 is both cubed and then square-rooted to derive our result. By applying exponent rules effectively, one can handle intricate expressions with ease.
Familiarity with exponents, including whole numbers and fractions, is essential for understanding and solving more complex mathematical problems seamlessly.