Problem 18 - Algebra and Trigonometry

Question

Evaluate each expression. (a) \(\left(\frac{1}{16}\right)^{-0.75}\) (b) \(0.25^{-0.5}\) (c) \(9^{1 / 3} \cdot 15^{1 / 3} \cdot 25^{1 / 3}\)

Step-by-Step Solution

Answer

(a) 8, (b) 2, (c) 15

Step 1: Understanding Negative Exponents

A negative exponent indicates that the base is inverted. The expression \(a^{-b}\) is equivalent to \(\frac{1}{a^b}\). For (a), \(\left(\frac{1}{16}\right)^{-0.75}\) becomes \(\left(\frac{1}{16}\right)^{0.75}\) with an inverted fraction, simplifying to \(16^{0.75}\).

Step 2: Applying Exponent Properties to Fractional Exponents

Fractional exponents mean taking roots. The expression \(16^{0.75}\) is equivalent to \((16^{3/4})\). Calculate the root first: \(16^{3/4} = (2^4)^{3/4} = 2^{3}\). Simplify to get \(8\).

Step 3: Simplifying Part (a)

From the previous steps, \((\frac{1}{16})^{-0.75} = 8\).

Step 4: Simplifying Negative Fractional Exponents in (b)

For (b), \(0.25^{-0.5}\). Rewrite as \(\frac{1}{0.25^{0.5}}\). Since \(0.25 = \frac{1}{4}\), find the square root: \((\frac{1}{4})^{0.5} = \frac{1}{2}\). Then invert the fraction to get \(2\).

Step 5: Simplifying Part (b)

Thus, \(0.25^{-0.5} = 2\).

Step 6: Using the Property of Fractional Exponents in (c)

For (c), simplify \(9^{1/3} \cdot 15^{1/3} \cdot 25^{1/3}\) using the property \(a^{1/3} \cdot b^{1/3} \cdot c^{1/3} = (abc)^{1/3}\). Compute \(9\cdot15\cdot25\) first.

Step 7: Calculating the Cube Root in (c)

Calculate the product: \(9 \cdot 15 \cdot 25 = 3375\). Find the cube root: \(3375^{1/3} = 15\).

Step 8: Simplifying Part (c)

The expression \(9^{1/3} \cdot 15^{1/3} \cdot 25^{1/3} = 15\).

Key Concepts

Understanding Negative Exponents

Negative exponents can seem tricky at first, but they actually have a simple rule. When you see a negative exponent, think of it as a way to "flip" the base into a fraction. If you have an expression like \(a^{-b}\), it is equal to \(\frac{1}{a^b}\). This means you take the reciprocal of the base raised to the positive exponent. This concept comes from the balancing nature of exponents. Every time you multiply by a certain number, a negative exponent undoes that multiplication by turning it into division. Thus, \(4^{-2}\) becomes \(\frac{1}{4^2} = \frac{1}{16}\). Likewise, \((\frac{1}{16})^{-0.75}\) can be rewritten simply by flipping the fraction, leading to \(16^{0.75}\). Understanding this concept can simplify your calculations considerably.

Understanding Fractional Exponents

Fractional exponents combine the operations of rooting and exponentiation. When you see a fractional exponent, such as \(a^{m/n}\), interpret it as combining two operations:

• The numerator \(m\) indicates that you raise the base to this power.
• The denominator \(n\) suggests you take the nth root of the base.

So, \(16^{3/4}\) means you should first take the 4th root of 16, which is 2, and then raise it to the power of 3, resulting in 8. This method simplifies complex calculations by breaking them down into smaller, more manageable parts. Applying these steps ensures clarity and accuracy in your mathematical solutions.

Simplifying Expressions Efficiently

Simplifying expressions involves transforming them into simpler or more digestible forms without changing their value. For example, turning \(0.25^{-0.5}\) into a simpler form involves using the rules of exponents and roots. Recognize that 0.25 is the same as \(\frac{1}{4}\), and that \(0.5\) as an exponent is equivalent to taking the square root. Hence, you perform \((\frac{1}{4})^{0.5} = \frac{1}{2}\), then invert it due to the negative exponent: \(\frac{1}{\frac{1}{2}} = 2\). This simplification process relies heavily on recognizing patterns and applying mathematical properties consistently.

Mastering Exponent Properties

Understanding exponent properties is crucial for simplifying expressions proficiently. Important properties include:

• Product of powers: \(a^m \cdot a^n = a^{m+n}\)
• Power of a power: \((a^m)^n = a^{m \cdot n}\)
• Power of a product: \((ab)^m = a^m \cdot b^m\)

These rules are useful for breaking down complex exponential expressions. For instance, in simplifying \(9^{1/3} \cdot 15^{1/3} \cdot 25^{1/3}\), recognize that multiplying terms raised to the same fractional power can be combined using the rule of fractional exponents. Hence, you multiply the bases 9, 15, and 25 first, then take the cube root of their product: \((9 \cdot 15 \cdot 25)^{1/3} = 15\). This method turns a seemingly complicated expression into something manageable through the application of fundamental exponent properties.