Problem 5 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without radicals or exponents.. $$ 9^{1 / 2} $$

Step-by-Step Solution

Answer

The simplified expression without radicals or exponents is \(3\).

Step 1: Understand the Properties of Exponents and Radicals

The given expression is in the form of \( 9^{\frac{1}{2}} \). Recall that the exponent represents a "power" or "root": an exponent of 2 means "square", an exponent of 3 means "cube", and so on. In our case, we have an exponent of \(\frac{1}{2}\), which means we are looking for the "square root". In general, when we have an expression in the form of \( x^{\frac{1}{n}} \), we are looking for the nth root of x.

Step 2: Find the Square Root

Now that we know we are looking for the square root of 9, we have to figure out which number, when multiplied by itself, will give us 9. We can easily see that the number 3 fulfills this requirement, as \(3 \times 3 = 9\). Therefore, the square root of 9 is 3, i.e., \(9^{\frac{1}{2}} = 3\).

Step 3: Rewrite the Expression without Radicals or Exponents

Now that we have simplified the expression by finding the square root of 9, we can rewrite the expression without using any radicals or exponents. So, \( 9^{\frac{1}{2}} = 3 \). The given expression has been rewritten without radicals or exponents.

Key Concepts

Understanding Radicals

In algebra, radicals are symbols used to represent roots of numbers. The most common radical is the square root, which you've probably seen as the radical sign (√). When you encounter a number like $$ \sqrt{9} $$\(\sqrt{9}\), it asks which number, multiplied by itself, equals 9. Radicals are helpful for solving equations where you want to isolate a variable by undoing exponents. They're like doing addition to undo subtraction, but for powers!

• A radical with a small 2 is a square root.
• A radical with a small 3 is a cube root.

Radicals can also be expressed with exponents, like $$ x^{\frac{1}{n}} $$\(x^{\frac{1}{n}}\). This expression shows us nth roots using a fractional exponent, like $$ \frac{1}{2} $$\(\frac{1}{2}\) for square roots. The interconnectedness of radicals and exponents makes it easy to transition between both forms, which can simplify complex math problems.

Exploring Exponents

Exponents are a powerful concept in algebra, often simplifying multiplication. They tell you how many times to multiply a number, called the "base", by itself. For example, $$ 2^3 \text{, meaning}\, 2 \times 2 \times 2 \text{, equals } 8. $$Our original expression, $$ 9^{\frac{1}{2}} $$,suggests the number 9 is involved in a root operation.Here's a breakdown of some common exponent properties:

• Multiplication becomes easier using the rule \( a^m \times a^n = a^{m+n} \).
• Division follows similar logic: \( \frac{a^m}{a^n} = a^{m-n} \).
• A power of zero results in one, \( a^0 = 1 \), as long as \( a eq 0 \).

Fractional exponents connect directly to radicals, translating an expression like $$ x^{\frac{1}{n}} $$into a more understandable root operation. Thus, understanding both forms enriches your problem-solving toolkit.

Demystifying Square Roots

The square root is one of the most fundamental concepts in mathematics. Essentially, it's the reverse of squaring a number. In our example,$$ 9^{\frac{1}{2}} = \sqrt{9} $$,meaning which number squared gives you 9.

• Every positive real number has two square roots: one positive and one negative. For 9, it’s 3 and -3.
• When you see $$ \sqrt{x} $$,it implies the principal (positive) square root.
• Squaring a number and taking its square root gets you back to your starting number: $$ \text{If } x = 3 \text{, then } (\sqrt{x})^2 = x \text{, or } 3^2 = 9. $$

By representing the expression $$ 9^{\frac{1}{2}} $$as 3, we've translated the concept of roots into a simple number, empowering us to solve equations efficiently without radicals or exponents.