Problem 88 - Algebra for College Students

Question

Use a calculator to simplify each of the following numerical expressions. Express your answers to the nearest hundredth. (a) \(\left(2^{-3}+3^{-3}\right)^{-2}\) (b) \(\left(4^{-3}-2^{-1}\right)^{-2}\) (c) \(\left(5^{-3}-3^{-5}\right)^{-1}\) (d) \(\left(6^{-2}+7^{-4}\right)^{-2}\) (e) \(\left(7^{-3}-2^{-4}\right)^{-2}\) (f) \(\left(3^{-4}+2^{-3}\right)^{-3}\)

Step-by-Step Solution

Answer

(a) 38.27, (b) 4.26, (c) 257.44, (d) 1257.95, (e) 281.56, (f) 392.47.

Step 1: Simplify the Base inside Parentheses for (a)

First, calculate each part: - Calculate \(2^{-3}\): This is \(\frac{1}{2^3} = \frac{1}{8} = 0.125\).- Calculate \(3^{-3}\): This is \(\frac{1}{3^3} = \frac{1}{27} \approx 0.037\).Now sum these values: \(0.125 + 0.037 = 0.162\).

Step 2: Raise to Power for (a)

Now raise the sum to the power of \(-2\): \(\left(0.162\right)^{-2} = \frac{1}{(0.162)^2} \approx 38.27\).This is the simplified value for part (a).

Step 3: Simplify the Base for (b)

First, break down each term: - Calculate \(4^{-3}\): This is \(\frac{1}{4^3} = \frac{1}{64} \approx 0.015625\).- Calculate \(2^{-1}\): This is \(\frac{1}{2} = 0.5\).Now, perform the subtraction: \(0.015625 - 0.5 = -0.484375\).

Step 4: Raise to Power for (b)

Raise the difference to the power of \(-2\): \((-0.484375)^{-2} = \frac{1}{(-0.484375)^2} \approx 4.26\).This is the simplified value for part (b).

Step 5: Simplify the Base for (c)

Calculate each term:- Calculate \(5^{-3}\): This is \(\frac{1}{5^3} = \frac{1}{125} = 0.008\).- Calculate \(3^{-5}\): This is \(\frac{1}{243} \approx 0.004115\).Subtract the values: \(0.008 - 0.004115 = 0.003885\).

Step 6: Raise to Power for (c)

Raise the difference to the power of \(-1\): \((0.003885)^{-1} = \frac{1}{0.003885} \approx 257.44\).This is the simplified value for part (c).

Step 7: Simplify the Base for (d)

Calculate each term:- Calculate \(6^{-2}\): This is \(\frac{1}{36} \approx 0.027778\).- Calculate \(7^{-4}\): This is \(\frac{1}{2401} \approx 0.000416\).Add the values: \(0.027778 + 0.000416 = 0.028194\).

Step 8: Raise to Power for (d)

Raise the sum to the power of \(-2\): \((0.028194)^{-2} = \frac{1}{(0.028194)^2} \approx 1257.95\).This is the simplified value for part (d).

Step 9: Simplify the Base for (e)

Calculate each term:- Calculate \(7^{-3}\): This is \(\frac{1}{343} \approx 0.00292\).- Calculate \(2^{-4}\): This is \(\frac{1}{16} = 0.0625\).Subtract the values: \(0.00292 - 0.0625 = -0.05958\).

Step 10: Raise to Power for (e)

Raise the difference to the power of \(-2\): \((-0.05958)^{-2} = \frac{1}{(-0.05958)^2} \approx 281.56\).This is the simplified value for part (e).

Step 11: Simplify the Base for (f)

Calculate each term:- Calculate \(3^{-4}\): This is \(\frac{1}{81} \approx 0.012346\).- Calculate \(2^{-3}\): This is \(\frac{1}{8} = 0.125\).Add the values: \(0.012346 + 0.125 = 0.137346\).

Step 12: Raise to Power for (f)

Raise the sum to the power of \(-3\): \((0.137346)^{-3} = \frac{1}{(0.137346)^3} \approx 392.47\).This is the simplified value for part (f).

Key Concepts

Negative Exponents

When dealing with negative exponents, the concept may initially seem intimidating. However, understanding it can be quite straightforward. A negative exponent indicates that the base should be taken as a reciprocal. For example, if you have a base of \( a^{-n} \), this can be expressed as \( \frac{1}{a^n} \).

• Convert \( x^{-3} \) into \( \frac{1}{x^3} \).
• Thus, \( 4^{-2} \) becomes \( \frac{1}{16} \).

Negative exponents essentially shift the number to the denominator of a fraction, allowing you to compute its reciprocal value easily. Whenever handling algebraic expressions with negative exponents, simply move the term containing the negative exponent across the fraction line and invert its sign.

Simplifying Numerical Expressions

Simplifying numerical expressions is about reducing complex algebraic equations or arithmetic operations to their simplest form. This involves performing calculations in the correct sequence, following the order of operations, which is typically denoted by PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
When faced with a problem like \((2^{-3}+3^{-3})^{-2}\), do the following:

• Solve the expressions with exponents first: Calculate \(2^{-3}\) and \(3^{-3}\).
• Add or subtract these intermediate results.
• Raise the parenthetical result to the intended power.

Calculating these accurately requires systematic handling of each part. It may be helpful to use a calculator for precise arithmetic, especially when decimals are involved.

Raising Numbers to a Power

Raising a number to a power involves multiplying the number by itself as many times as the exponent specifies. For example, "cubing" a number means raising it to the power of three, like \( 2^3 = 2 \times 2 \times 2 = 8 \).
For negative bases, be cautious with exponent evenness or oddness:

• \((-2)^3 = -8\) since multiplying a negative by itself an odd number of times remains negative.
• \((-2)^2 = 4\) since an even exponent results in a positive number.

When simplifying expressions, ensure each part is calculated separately, then combined. A calculator can be very handy, especially with non-integer exponents or complex decimal results.

Step-by-Step Problem Solving

Solving algebraic problems in a step-by-step fashion allows you to break down complex tasks into manageable parts. Each step resolves one piece of the problem, making errors less likely and calculations easier to control.
Here's how to tackle a problem step-by-step:

• Isolate the expression you need to simplify.
• Handle operations inside parentheses first.
• Compute individual exponentiations or multiplications.
• Follow with additions, subtractions, or any other operations required.

As each piece is addressed, re-evaluate the expression to simplify further. Utilizing a methodical approach ensures clarity and accuracy, especially for complex numerical expressions.