Problem 2 - Elementary and Intermediate Algebra

Question

Fill in the blanks. We read \(16^{3 / 2}\) as " 16 to the three- _____ power."

Step-by-Step Solution

Answer

halves

Step 1: Understanding the Exponentiation Expression

The expression given is \(16^{3/2}\). This indicates that the base, 16, is raised to the power of the fraction \(\frac{3}{2}\). Exponents can be expressed as any real number, including fractions and decimals.

Step 2: Breaking Down the Fractional Exponent

In mathematics, a fractional exponent, such as \(\frac{3}{2}\), can be interpreted in terms of roots and powers. Specifically, the denominator 2 of \(\frac{3}{2}\) suggests taking the square root, and the numerator 3 means raising to the power of 3.

Step 3: Simplifying the Interpretation

To simplify, when we talk about \(16^{3/2}\), we can say it's '16 raised to the three-halves'. Here, "halves" refers to the denominator (2) of the fraction \(\frac{3}{2}\), which corresponds to taking the square root.

Step 4: Fill in the Blank

Now that we have explored the breakdown of the expression, we can fill in the blank with the term that describes the denominator of the fraction: "halves." Thus, \(16^{3/2}\) is read as '16 to the three-halves power.'

Key Concepts

Exponents

Exponents are powerful tools in mathematics used to express repeated multiplication. When you see a number like \(2^3\), it means that the base, 2, is multiplied by itself 3 times, resulting in 8. Exponents tell you how many times to use the base in a multiplication. Understanding how exponents work helps make complex calculations easier and helps in simplifying large numbers.
For example, in the expression \(16^{3/2}\), the base is 16, and it is raised to the power of \(\frac{3}{2}\). This not only aids in simplification but also introduces us to the world of fractional exponents, which can involve both roots and further multiplication steps.

• An exponent like 3 indicates cube, or multiplying the base by itself twice more after introducing the base.
• Exponents can be any real number, making them versatile tools in mathematics.

Square Root

The square root is an essential concept in mathematics, especially when dealing with exponents. Finding the square root of a number is like asking: "What number, when multiplied by itself, gives me this result?" For instance, the square root of 16 is 4 because \(4 \times 4 = 16\).
Square roots are frequently noted when dealing with fractional exponents where the denominator is 2. In the example of \(16^{3/2}\), the '2' in the denominator indicates taking a square root.

• The notation is \(\sqrt{}\), and the number inside is the 'radicand' which we wish to find the root of.
• Taking the square root is the inverse operation of squaring a number.

Comprehending square roots as part of fractional exponents enables smoother transitions between radical and exponential forms.

Mathematical Expressions

Mathematical expressions consist of numbers, variables, and operations organized in a meaningful way to represent a value or relationship. These are fundamental to expressing and solving equations and can involve several mathematical concepts.
In expressions with fractional exponents, like \(16^{3/2}\), it combines the idea of an exponent with root extraction. The expression showcases how numbers can be manipulated to simplify or alter their form.

• Every expression has components like bases, exponents, or operations such as addition or multiplication.
• The organization of expressions matters, as it dictates the order of operations or understanding.
• Mathematical expressions can handle complex operations through symbolic simplification.

Deconstructing mathematical expressions into parts helps in better understanding the operations involved and how to perform them.