Problem 66 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. See Example 7. $$ 49^{-1 / 2} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{7}\).

Step 1: Understand the Expression

The expression given is \(49^{-1/2}\). We need to simplify this expression involving an exponent that is both negative and a fraction.

Step 2: Apply the Negative Exponent Rule

Recall the rule for negative exponents: \(a^{-n} = \frac{1}{a^n}\). Applying this rule to \(49^{-1/2}\), we get:\[49^{-1/2} = \frac{1}{49^{1/2}}\]

Step 3: Apply the Fractional Exponent Rule

A fractional exponent \(a^{1/n}\) is equivalent to the \(n\)-th root of \(a\). So, \(49^{1/2}\) is the square root of \(49\). This gives us:\[49^{1/2} = \sqrt{49}\]

Step 4: Evaluate the Square Root

Calculate the square root of \(49\). Since \(49\) is a perfect square, \(\sqrt{49} = 7\).

Step 5: Final Simplification

Plug the result from the previous step back into the fraction we derived earlier:\[49^{-1/2} = \frac{1}{49^{1/2}} = \frac{1}{7}\]

Key Concepts

Understanding Negative Exponents

Negative exponents can initially seem confusing, but they're quite straightforward once understood. Imagine you're moving backwards in terms of multiplication.
If you have an expression like \(a^{-n}\), this is equivalent to \(\frac{1}{a^n}\).
The negative sign in the exponent doesn't mean the result is negative; it indicates you take the reciprocal instead.

• This concept is essentially about flipping the fraction. For example, \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\).
• Any non-zero number raised to the power of zero is 1, but negative exponents will give you a fraction instead.
• In our specific exercise, we applied this to \(49^{-1/2}\) resulting in \(\frac{1}{49^{1/2}}\).

Understanding negative exponents helps simplify expressions and solve various kinds of algebraic problems easily.

Comprehending Fractional Exponents

Fractional exponents are simply another way of expressing roots, which might otherwise seem less intuitive at first glance.
When you see an exponent like \(a^{1/n}\), this means you're taking the \(n\)-th root of \(a\).

• This changes the base into a root form. For example, \(8^{1/3}\) becomes \(\sqrt[3]{8}\), which evaluates to 2.
• Any number raised to a power represented by a fraction with a numerator of 1 is still a root. For instance, \(27^{1/3} = 3\), because 3 cubed equals 27.
• In the original exercise, we applied this by transforming \(49^{1/2}\) into \(\sqrt{49}\), which simplified further to 7.

Fractional exponents follow the same rules as standard exponents, which makes calculations consistent and manageable.

Simplifying with Square Roots

Square roots are one of the simplest forms of fractional exponents and often appear in algebraic problems.
They essentially ask "what number, when multiplied by itself, gives the original number?"

• For perfect squares, this is easy to find. For instance, the square root of 16 is 4, because \(4 \times 4 = 16\).
• Calculating square roots is straightforward when dealing with perfect squares, but can become more complex with non-perfect squares, often requiring calculator use.
• In the exercise, we found that \(\sqrt{49}\) equaled 7, which was key to simplifying the overall expression \(\frac{1}{7}\).

Square roots help in breaking down more complex expressions into simpler forms, aiding in deriving solutions effectively.