Problem 13 - Intermediate Algebra

Question

For Problems \(1-30\), evaluate each numerical expression. $$ 4^{\frac{3}{2}} $$

Step-by-Step Solution

Answer

The expression \( 4^{\frac{3}{2}} \) evaluates to 8.

Step 1: Understand the Expression

The expression given is \( 4^{\frac{3}{2}} \). This means we need to evaluate the cube of the square root of \( 4 \).

Step 2: Find the Square Root

Calculate the square root of \( 4 \). We know that \( \sqrt{4} = 2 \) because \( 2 \times 2 = 4 \).

Step 3: Raise to the Third Power

Now that we have the square root, raise it to the third power: \( 2^3 \). Calculate \( 2^3 = 2 \times 2 \times 2 = 8 \).

Step 4: Conclusion

Thus, the expression \( 4^{\frac{3}{2}} \) evaluates to \( 8 \).

Key Concepts

Numerical Expressions

Numerical expressions are mathematical phrases that use numbers and operation symbols, such as addition, subtraction, multiplication, division, and exponentiation, to represent a value. These expressions simplify using established mathematical operations to find a single value.

For instance, the expression \( 4^{\frac{3}{2}} \) is a numerical expression involving exponentiation. To evaluate it, we must follow specific steps that involve understanding the meanings of the numbers and operations within the expression. By systematically breaking down the expression, as we did in the solution process, we can find its numerical value. This involves identifying the operations to perform, such as square roots and powers, which lead us to the final solution. Numerical expressions provide a convenient way to represent complex calculations succinctly, making them fundamental in algebra and higher-level mathematics.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the radical symbol \( \sqrt{} \), and it plays a crucial role in various mathematical computations and problem-solving scenarios.

For example, in the expression \( 4^{\frac{3}{2}} \), the fraction \( \frac{3}{2} \) indicates taking the square root as one of the operations. Finding the square root of 4 means we are looking for a number which, when squared, equals 4. The answer is 2, because \( 2 \times 2 = 4 \).

• The square root helps break down more complex expressions into simpler forms that can be more easily evaluated.
• Understanding how to calculate square roots is integral, especially when dealing with powers where the exponents are fractions.

The ability to find the square root efficiently is essential, not only in school algebra, but also in various scientific and engineering applications.

Powers

In mathematics, powers refer to the operation of raising a number or expression to a certain exponent. The process of exponentiation involves multiplying a number by itself a specific number of times, as indicated by the exponent.

For example, in the expression \( 4^{\frac{3}{2}} \), the fraction \( \frac{3}{2} \) represents both taking the square root of 4 (due to the 1/2 part) and then cubing the result, which is related to the 3 part. These steps demonstrate how different operations intertwine within a power expression.

When evaluating powers, follow these simple guidelines:

• Break down the exponent into manageable parts, such as taking the square root or cube root first, then raising to the integer power.
• Understand that fractional exponents are a combination of roots and powers. For example, \( 4^{\frac{3}{2}} \) involves the square root and then cubing the result (\( 2^3 \)).

Powers are fundamental in expressing repeated multiplication succinctly and are widely applied across different branches of mathematics and sciences, making them indispensable for solving real-world problems and mathematical theories.