Problem 24 - Algebra and Trigonometry

Question

\(15-24\) . Evaluate each expression. $$ \begin{array}{llll}{\text { (a) }\left(\frac{1}{16}\right)^{-0.75}} & {\text { (b) } 0.25^{-0.5}} & {\text { (c) } 9^{1 / 3} \cdot 15^{1 / 3} \cdot 25^{1 / 3}}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Simplify Exponent for Part (a)

To solve \( \left(\frac{1}{16}\right)^{-0.75} \), recognize that raising a base to a negative exponent means taking the reciprocal of the base raised to the positive exponent. Thus, \( \left(\frac{1}{16}\right)^{-0.75} = 16^{0.75} \).

Step 2: Convert the Exponent to a Fraction

The exponent \(0.75\) is equivalent to \(\frac{3}{4}\). Therefore, \(16^{0.75} = 16^{\frac{3}{4}}\).

Step 3: Evaluate the Radicals for Part (a)

\(16^{\frac{3}{4}}\) can be rewritten using radical notation. This means taking the fourth root of 16 and then raising the result to the power of 3. Since \(\sqrt[4]{16} = 2\), we have \(2^3 = 8\).

Step 4: Evaluate Part (b)

For \(0.25^{-0.5}\), convert \(0.25\) to a fraction, which is \(\frac{1}{4}\). Thus, \(0.25^{-0.5} = \left(\frac{1}{4}\right)^{-0.5} = 4^{0.5}\), which is the square root of 4, so \(\sqrt{4} = 2\).

Step 5: Simplify and Compute Part (c)

For \(9^{1/3} \cdot 15^{1/3} \cdot 25^{1/3}\), recognize that when bases are multiplied, their exponents can be summed if they are raised to the same power. Thus, this is equivalent to \((9 \cdot 15 \cdot 25)^{1/3}\).

Step 6: Multiply Bases for Part (c) and Find Cube Root

Calculate \(9 \cdot 15 \cdot 25\), which equals 3375. Then, find the cube root: \( \sqrt[3]{3375}\). Solving gives \(\sqrt[3]{3375} = 15\).

Key Concepts

Negative Exponents

Negative exponents might seem confusing initially, but they can be understood quite simply. A negative exponent indicates that you take the reciprocal of the base and then apply the positive exponent. For example, in the expression \( \left(\frac{1}{16}\right)^{-0.75} \), the negative exponent of \(-0.75\) means you need to take the reciprocal of \( \frac{1}{16} \) to make the exponent positive. So, \( \left(\frac{1}{16}\right)^{-0.75} \) becomes \( 16^{0.75} \).
This rule simplifies calculations and helps in understanding how exponents relate to division. Some tips for working with negative exponents:

• Always take the reciprocal of the base to eliminate the negative sign in the exponent.
• Remember that \( a^{-b} = \frac{1}{a^{b}} \).
• Use this concept to simplify and solve expressions with negative exponents easily.

Fractional Exponents

Fractional exponents are another way to represent roots and powers in a compact form. They blend two operations:

• The numerator is the power, and it suggests how many times to multiply the number by itself.
• The denominator is the root and indicates the degree of the root to be taken.

For example, the expression \( 16^{0.75} \) can be broken down as \( 16^{\frac{3}{4}} \), where 3 is the power and 4 is the root. This suggests taking the fourth root of 16 and then raising the outcome to the power of 3. Thus:

• Find the fourth root of 16: \( \sqrt[4]{16} = 2 \).
• Then, raise 2 to the power of 3: \( 2^3 = 8 \).

Understanding fractional exponents enables a more straightforward approach to working with radical expressions and complex calculations.

Radical Notation

Radical notation is a way to represent roots, such as square roots or cube roots, which is often seen with a radical symbol (\( \sqrt{} \)). This notation connects seamlessly with fractional exponents.To convert between radical notation and fractional exponents, use the fraction's denominator as the index or root of the radical. For instance:

• \( 16^{\frac{3}{4}} \) indicates the use of both radical and power—meaning you first take the fourth root of 16, and then cube the result.
• It shows that \( 16^{0.75} \) can also be expressed as \( \sqrt[4]{16^3} \).

Using radical notation simplifies the process of handling complex expressions and helps visualize the operations. By mastering the concept of radical notation, you can work confidently with expressions involving roots and powers.