Problem 48 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(-\frac{1}{8}\right)^{-5 / 3} $$

Step-by-Step Solution

Answer

The expression evaluates to \(-32\).

Step 1: Understand negative exponents

The expression \( \left(-\frac{1}{8}\right)^{-5/3} \) contains a negative exponent. A negative exponent indicates that you have to take the reciprocal of the base and change the sign of the exponent. Therefore, \( a^{-b} = \frac{1}{a^b} \). Apply this rule to rewrite the expression.

Step 2: Rewrite with reciprocal

Rewrite the original expression \( \left(-\frac{1}{8}\right)^{-5/3} \) as \( \left(-\frac{1}{8}\right)^{5/3} \), which means to find the reciprocal, or \( \left(-\frac{8}{1}\right)^{5/3} \). This simplifies to \( (-8)^{5/3} \).

Step 3: Break down the exponent

The exponent \( \frac{5}{3} \) implies two operations: raising the number to the power of 5 and then taking the cube root (or vice versa). It can be rewritten as \( ((-8)^{1/3})^5 \).

Step 4: Calculate cube root

Find the cube root of \(-8\). The cube root of a number \( x \) is a number \( y \) such that \( y^3 = x \). Since \( (-2)^3 = -8 \), the cube root of \(-8\) is \(-2\).

Step 5: Raise to fifth power

Take the result from the previous step and raise it to the fifth power: \((-2)^5\). Calculate \((-2)^5\) by multiplying \(-2\) five times: \((-2) \times (-2) \times (-2) \times (-2) \times (-2) = -32\). Thus, the final result is \(-32\).

Key Concepts

Negative Exponents

Negative exponents might sound a bit tricky at first, but they are simply a way of expressing the reciprocal of a number raised to a certain power. When you see a negative exponent, it essentially asks you to take the reciprocal of the base number and then use the positive version of the exponent. For instance, if you have an expression like \(a^{-b}\), this is equivalent to \(\frac{1}{a^b}\). This means you're flipping the number (finding the reciprocal) and then computing the power.

• Always flip the base when you see a negative exponent.
• Change the sign of the exponent to positive.

Once you have this concept down, you can apply it to simplifying various algebraic expressions, just as we did with \( \left(-\frac{1}{8}\right)^{-5 / 3} \). By understanding negative exponents, complex-looking expressions become much simpler to handle.

Reciprocal

Taking the reciprocal of a number means flipping it upside down. It is essential when dealing with negative exponents because it helps transform and simplify expressions.
For example, the reciprocal of \( \frac{1}{8} \) is \( 8 \) and the reciprocal of \( -\frac{1}{8} \) is \(-8\).

• For any fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
• The reciprocal of a whole number \( a \) is \( \frac{1}{a} \).

Having a solid grasp of this concept is crucial because it not only aids in understanding how to handle negative exponents but also simplifies solving equations and manipulating fractions. In our specific problem, converting \(\left(-\frac{1}{8}\right)\) to \(-8\) made it much easier to further handle the exponent operations.

Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. This is a key operation when dealing with fractional exponents. The cube root is denoted by the radical symbol with a small three (\( \sqrt[3]{x} \)).
For instance, when we take the cube root of \(-8\), we ask ourselves, "What number multiplied by itself three times equals -8?" This number is \(-2\), since \((-2)^3 = -2 \times -2 \times -2 = -8\).

• The cube root is particularly useful for simplifying expressions with fractional exponents.
• Remember that unlike square roots, cube roots can be of negative numbers because a negative number times itself three times remains negative.

Knowing how to manage cube roots helps in smoothly breaking down problems involving exponents, simplifying them in the process.

Power of a Number

Raising a number to a power means multiplying the number by itself a certain number of times. The power of a number is expressed using an exponent.
For example, in \((-2)^5\), \(-2\) is the base and 5 is the exponent. This expression tells us to multiply \(-2\) by itself five times: \((-2) \times (-2) \times (-2) \times (-2) \times (-2) = -32\).

• Powers are a fundamental operation in algebra, used across various topics.
• An even power of a negative number results in a positive product, while an odd power remains negative.

Understanding how to compute powers of numbers is crucial, especially for handling more complex calculations involving exponents, as seen in the evaluation of expressions with fractional and negative exponents.