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Magazine Chapter 1: Problem 42
Evaluate each expression without using a calculator. $$ (-27)^{-2 / 3} $$
Short Answer
Expert verified
The expression evaluates to \(\frac{1}{9}\).
Step by step solution
01
Understand the Expression
We need to evaluate the expression \((-27)^{-2/3}\). The exponent \(-2/3\) consists of a negative sign and a fraction, indicating both an inverse operation and a root operation.
02
Address the Negative Exponent
The negative exponent \(-2/3\) means we need to take the reciprocal of the base raised to the positive exponent. Thus, \((-27)^{-2/3} = \frac{1}{(-27)^{2/3}}\).
03
Simplify the Fractional Exponent
The fractional exponent \(2/3\) indicates that we first take the cube root and then square the result. Thus, \((-27)^{2/3} = ((-27)^{1/3})^2\).
04
Calculate the Cube Root
Calculate \((-27)^{1/3}\), which is the cube root of -27. The cube root of -27 is -3, because \((-3) imes (-3) imes (-3) = -27\).
05
Square the Result
Now square the cube root result: \((-3)^2 = 9\). This means \((-27)^{2/3} = 9\).
06
Evaluate the Final Expression
Returning to the expression, we have \(\frac{1}{9}\) because \((-27)^{-2/3} = \frac{1}{(-27)^{2/3}} = \frac{1}{9}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Negative Exponents
Negative exponents can sometimes seem intimidating, but they're quite straightforward once you learn the rules. When you see a negative exponent like in \((-27)^{-2/3}\), think of it as an instruction to flip the number - to find the reciprocal.
A reciprocal simply means one over the number.
- For example, if you have \(a^{-n}\), it means \(\frac{1}{a^n}\).
So, in the exercise, \((-27)^{-2/3}\) becomes \(\frac{1}{(-27)^{2/3}}\).
This step is crucial because it transforms the problem from dealing with negative powers to handling positive powers, making it easier to manage further calculations.
A reciprocal simply means one over the number.
- For example, if you have \(a^{-n}\), it means \(\frac{1}{a^n}\).
So, in the exercise, \((-27)^{-2/3}\) becomes \(\frac{1}{(-27)^{2/3}}\).
This step is crucial because it transforms the problem from dealing with negative powers to handling positive powers, making it easier to manage further calculations.
Reciprocal
The reciprocal of a number is simply that number flipped.
For example, the reciprocal of 5 is \(\frac{1}{5}\).
When working with exponents, understanding reciprocals helps when you've got negative exponents.
This is because a negative exponent directly indicates using a reciprocal.
Consider the expression \((-27)^{-2/3}\). Here we needed the reciprocal due to the negative exponent:
- Thus, \((-27)^{-2/3} = \frac{1}{(-27)^{2/3}}\).
Now, instead of focusing on a negative exponent, we shift our attention to solving \(\frac{1}{(-27)^{2/3}}\) with positive exponents.
This technique makes calculations easier and often tidier.
For example, the reciprocal of 5 is \(\frac{1}{5}\).
When working with exponents, understanding reciprocals helps when you've got negative exponents.
This is because a negative exponent directly indicates using a reciprocal.
Consider the expression \((-27)^{-2/3}\). Here we needed the reciprocal due to the negative exponent:
- Thus, \((-27)^{-2/3} = \frac{1}{(-27)^{2/3}}\).
Now, instead of focusing on a negative exponent, we shift our attention to solving \(\frac{1}{(-27)^{2/3}}\) with positive exponents.
This technique makes calculations easier and often tidier.
Cube Root
Calculating a cube root may sound complex, but it's simpler than it seems.
The cube root of a number is another number which, when multiplied by itself three times, gives the original number.
In our case for \((-27)\), we looked for what number multiplied three times results in \(-27\).
- It turns out this number is \(-3\) since \((-3) \times (-3) \times (-3) = -27\).
Finding cube roots acts as the first step in simplifying fractional exponents like \(\frac{2}{3}\).
In the expression \((-27)^{2/3}\), the cube root is calculated first, leading to the next step of handling the exponent.
The cube root of a number is another number which, when multiplied by itself three times, gives the original number.
In our case for \((-27)\), we looked for what number multiplied three times results in \(-27\).
- It turns out this number is \(-3\) since \((-3) \times (-3) \times (-3) = -27\).
Finding cube roots acts as the first step in simplifying fractional exponents like \(\frac{2}{3}\).
In the expression \((-27)^{2/3}\), the cube root is calculated first, leading to the next step of handling the exponent.
Square
Squaring a number means multiplying it by itself.
Once we found the cube root, which is \(-3\), the exercise required us to square \(-3\) to move on with the calculation.
- When you square \(-3\), the negative sign is also factored in: \((-3) \times (-3) = 9\).
This step often concludes operations with fractional exponents like \(\frac{2}{3}\), where the second part of the calculation is squaring the initial result.
It brings our answer one step closer to completion, allowing us then to find the reciprocal for a final answer if necessary.
Once we found the cube root, which is \(-3\), the exercise required us to square \(-3\) to move on with the calculation.
- When you square \(-3\), the negative sign is also factored in: \((-3) \times (-3) = 9\).
This step often concludes operations with fractional exponents like \(\frac{2}{3}\), where the second part of the calculation is squaring the initial result.
It brings our answer one step closer to completion, allowing us then to find the reciprocal for a final answer if necessary.
