Problem 4 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\(4^{3/2} = 8\) using either method.

Step 1: Understanding the Exponential Expression

The expression \(4^{3/2}\) is a power expression where 4 is the base and \(\frac{3}{2}\) is the exponent. This means you are raising 4 to the power of \(\frac{3}{2}\). This exponent is a rational number, and we can interpret it in two different ways: as a square root followed by a cube, or as a cube followed by a square root.

Step 2: First Method: Apply the Square Root First

According to this method, we first take the square root of 4, because the denominator of the exponent \(\frac{3}{2}\) is 2. The square root of 4 is:\[4^{1/2} = \sqrt{4} = 2.\]Then, raise this result to the power of 3 (numerator of the exponent):\[2^3 = 8.\]

Step 3: Second Method: Apply the Cube First

In this method, we first compute \(4^3\):\[4^3 = 4 \times 4 \times 4 = 64.\]Then, take the square root of this result, because the denominator of the exponent is 2:\[\sqrt{64} = 8.\]

Step 4: Conclusion of Both Methods

Both methods yield the same result. Therefore, \(4^{3/2} = 8\). This shows the flexibility of applying different operations of the exponent in the sequence depending on convenience.

Key Concepts

Rational Exponents

Rational exponents are a fascinating concept where we extend the idea of exponents from whole numbers to fractions. When you see an exponent like \( \frac{3}{2} \), it's called a rational exponent because it's a ratio or fraction. This exponent can be thought of as two operations happening in sequence: the denominator indicates a root, and the numerator signifies a power.
For example, in the expression \( 4^{3/2} \), the 2 in the denominator suggests taking a square root, while the 3 in the numerator tells us to cube the result. This dual nature allows for flexibility in computation, offering multiple ways to arrive at the same answer.
Understanding rational exponents allows you to rewrite expressions involving roots and powers in a more unified and elegant form, which can simplify computation and help in higher math problem-solving.

Exponentiation Methods

Exponentiation is the process of raising a number, known as the base, to a power or exponent. It is one of the fundamental operations in mathematics. There are various methods to approach exponentiation, especially when dealing with rational exponents.
You can solve an expression in different steps based on the exponent's denominator and numerator. For example:

• When the denominator suggests a root, you can take that root first and then apply the power indicated by the numerator. This approach is sometimes more intuitive.
• Alternatively, you may choose to apply the power first and then take the root. This method can be useful when the base is a neat number whose higher powers are simple.

Understanding and choosing between these methods can make calculations quicker and ensure you handle complex expressions correctly.

Square Roots

The square root is a specific type of root where you are looking for a number that, when multiplied by itself, will give you the original number. It is denoted by the radical symbol \( \sqrt{} \). Calculating the square root is a common step when dealing with rational exponents that have a denominator of 2, like \( 4^{1/2} \) which equals \( \sqrt{4} = 2 \).
Square roots are crucial in various mathematical contexts, including simplification and solving quadratic equations. Knowing the square roots of perfect squares, such as 1, 4, 9, 16, up to 100 and beyond, can greatly assist in performing calculations more quickly and accurately. They also play a vital role in geometry, particularly in calculating distances through the Pythagorean theorem.

Cubic Powers

Cubic powers involve raising a number to the power of three. This means multiplying the number by itself twice more. For example, \( 4^3 \) equals \( 4 \times 4 \times 4 = 64 \). Cubic powers appear frequently in calculations related to volume, given how they relate to three-dimensional measurements.
When working with rational exponents, cubic powers offer a stepping stone for further operations, such as taking roots. This two-step process of cubing followed by a square root calculation, or vice versa, helps to break down complex expressions into manageable parts.
Understanding these powers can also aid in recognizing and factoring polynomials, especially those in the form of perfect cubes, contributing to a broader comprehension of algebraic expressions and equations.