Problem 16 - Algebra and Trigonometry

Question

Evaluate each expression. (a) \((-1000)^{-2 / 3}\) (b) \(10,000^{-3 / 2}\) (c) \((-8000)^{4 / 3}\)

Step-by-Step Solution

Answer

(a) 0.01 (b) 0.000001 (c) 160,000

Step 1: Understanding Negative Exponents

For negative exponents, recall that \(a^{-n} = \frac{1}{a^n}\). This means we need to reciprocate the base and then apply the positive exponent.

Step 2: Simplifying (a)

We have \((-1000)^{-2 / 3}\).

Step 3: Substep 2.1: Convert to Positive Exponent

Using the rule for negative exponents, we convert it to: \[ (-1000)^{-2/3} = \frac{1}{(-1000)^{2/3}} \]

Step 4: Substep 2.2: Calculate the Denominator

Find \((-1000)^{2/3}\). First, calculate the cube root: \( \sqrt[3]{-1000} = -10\). Then, square it: \((-10)^2 = 100\). So, \((-1000)^{2/3} = 100\).

Step 5: Substep 2.3: Final Calculation

Putting it all together:\[ (-1000)^{-2/3} = \frac{1}{100} = 0.01 \]

Step 3: Simplifying (b)

Now for \(10,000^{-3/2}\).

Step 7: Substep 3.1: Convert to Positive Exponent

Convert the expression using the rule: \[ 10,000^{-3/2} = \frac{1}{(10,000)^{3/2}} \]

Step 8: Substep 3.2: Calculate the Denominator

Find \((10,000)^{3/2}\).First, calculate the square root: \( \sqrt{10,000} = 100\).Then, cube it: \(100^3 = 1,000,000\). So, \((10,000)^{3/2} = 1,000,000\).

Step 9: Substep 3.3: Final Calculation

Putting it all together:\[ 10,000^{-3/2} = \frac{1}{1,000,000} = 0.000001 \]

Step 4: Simplifying (c)

Now for \((-8000)^{4/3}\).

Step 11: Substep 4.1: Break Down the Exponentiation

First, find the cube root of -8000: \( \sqrt[3]{-8000} = -20 \).Then raise it to the power of 4: \((-20)^4 = 160,000\).

Step 12: Substep 4.2: Final Calculation

Thus, \((-8000)^{4/3} = 160,000\).

Key Concepts

Negative Exponents

Negative exponents might feel intimidating at first, but they are actually quite simple. The rule of negative exponents is: \(a^{-n} = \frac{1}{a^n}\). This means that instead of having a negative exponent, you "flip" the base to the denominator and make the exponent positive.
For example, if you have \(10^{-3}\), you can rewrite it as \(\frac{1}{10^3}\). So, dealing with negative exponents is just about taking the reciprocal of the base and working with a positive exponent instead.

• Flip the base to turn the exponent positive.
• Simplify using the reciprocal approach.

It's a handy tool for breaking down complex expressions and simplifies calculations.

Cube Roots

Cube roots find the number that, when multiplied three times by itself, equals the given value. The cube root is represented by the symbol \(\sqrt[3]{a}\). For instance, the cube root of -1000 is -10, because \((-10) \times (-10) \times (-10) = -1000\).
Cube roots are especially useful in algebra when dealing with expressions where fractional exponents are involved. To calculate a cube root, you can use various methods, including:

• Estimating the value.
• Using a calculator for accurate results.
• Breaking down numbers to their prime factors.

Understanding the cube root helps in simplifying expressions with roots and exponents, turning them from intimidating to manageable.

Square Roots

Just like cube roots, square roots help simplify expressions by finding a number that is multiplied by itself to get the given number. The square root is represented by the symbol \(\sqrt{a}\). For example, the square root of 10000 is 100, because \(100 \times 100 = 10000\).
Finding square roots is a common task in math, especially in quadratic equations and geometry. You might use:

• Perfect square recognition, if the number is a perfect square.
• A calculator for non-perfect squares.
• Trial and error or estimation methods for mental calculations.

Mastering square root calculations enhances your ability to deal with powers and roots in various math problems.

Reciprocal of a Number

The reciprocal of a number is easy to grasp once you think about it in simple terms. It's just "flipping" a fraction. For any non-zero number \(a\), its reciprocal is \(\frac{1}{a}\). So, the reciprocal of 2 is \(\frac{1}{2}\).
Understanding reciprocals is fundamental in solving equations, especially those involving divisions and simplifying fractions. You can apply this concept by:

• Recognizing fractions easily. For example, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).
• Using reciprocals to simplify division problems by turning them into multiplications.
• Ensuring the base is non-zero before finding the reciprocal.

Knowing how to find reciprocals lets you tackle inverse operations with confidence.

Power of a Number

The power of a number—or an exponent—indicates how many times the number (the base) is multiplied by itself. It's written as \(a^n\), where \(a\) is the base and \(n\) is the exponent. For example, in \(10^3\), 10 is the base and 3 is the exponent, which means \(10 \times 10 \times 10 = 1000\).
Powers are a shorthand way to deal with large multiplication operations and are essential in algebra and scientific calculations. Key components include:

• A base, which is the number being multiplied.
• An exponent, which shows how many times the base multiplies itself.
• Recognizing that any number raised to the power of 1 is itself, and to the power of 0 is 1 (except zero).

Grasping the concept of powers and how to apply them allows for easier simplification and computation in math problems.