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Chapter 7: Problem 76

Solve each equation. $$(w+3)^{-1 / 3}=\frac{1}{3}$$

Short Answer

Expert verified
w = 24

Step by step solution

01

Isolate the expression inside the exponent

Remove the negative exponent by raising both sides to the power of \(-3\): \[((w+3)^{-1/3})^{-3} = \frac{1}{3}^{-3}\] This simplifies to: \[w + 3 = 3^3\]
02

Calculate the right side

Simplify \[3^3\] on the right side: \[w + 3 = 27\]
03

Solve for w

Subtract 3 from both sides to solve for \w\: \[w + 3 - 3 = 27 - 3\] \[w = 24\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Negative Exponents
Negative exponents might seem tricky at first, but they are actually straightforward once you understand the rule. When you have a negative exponent, it indicates that you need to take the reciprocal (or the inverse) of the base and then raise it to the corresponding positive power. For example, \((a)^{-n} = \frac{1}{a^n}\).

  • If we have \((3)^{-2} \), it is the same as \((\frac{1}{3^2}) = \frac{1}{9}\).
  • Similarly, if \((x + 1)^{-3} = \frac{1}{(x+1)^3}\).
In our exercise, \((w + 3)^{-1/3}\) being equal to \({1}/{3}\) means we need to handle the exponent by making the exponent positive. Raising both sides to the power of -3 did just that by canceling out the exponent on the left side. After this step, the equation simplifies considerably.
Isolating Expressions
Isolating an expression is often a key step in solving equations. This means getting the variable we want to solve for by itself on one side of the equation.
  • In the exercise, we started by isolating the term with the exponent, \((w+3)^{-1/3}\).
To isolate, raise both sides of the equation to the same power (here, -3) which removes the exponent on the left side.
  • This is possible because raising a term to a power and then raising the result to the inverse of that power cancels the exponent, resulting in just the base.
This gave us \(w + 3 = 27\), much simpler to solve.
Exponentiation
Exponentiation refers to the operation of raising a number or expression to a power. In the equation, after isolating the expression we simplified by exponentiating both sides. In this instance:
  • The left side, \((w+3)\), remained because we raised to a power of 1 (canceling out the exponent).
  • The right side involved calculating \((3^3)\).
Here鈥檚 a quick breakdown of exponentiation steps: \((a^n) means multiplying a by itself n times. For example, 3^3 is 3 * 3 * 3 = 27.\)

  • After simplifying the exponentiation, we solved for \(w\) by isolating it through basic arithmetic operations: Subtract 3 from both sides to get the value of w.
This step-by-step process shows how understanding exponentiation can make even complex equations manageable.

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