Problem 60 - Algebra 2

Question

Simplify each number. $$-(-27)^{-\frac{4}{3}}$$

Step-by-Step Solution

Answer

The simplified version of the given number is \(1 / 81\).

Step 1: Understand negative exponent usage

In mathematics, a negative exponent indicates that we should turn the base into its reciprocal. Therefore, \( - (-27)^{-4/3} \) becomes \(1 / -(-27)^{4/3}\). However, since the base (-27) is also negative, this double negative will turn into a positive number, which simplifies our problem even further into \(1 / 27^{4/3}.\)

Step 2: Apply exponent rules

The exponent \(4/3\) can be understood as 'cube root and then raise to the 4th power'. Apply these operations to 27. First, calculate the cube root of 27, which is 3. Then, raise that result to the power of 4, which gives 81. This leaves us with \(1 / 81\).

Step 3: Simplify the result

Our final step is to simplify the fraction \(1 / 81\) as much as possible. However, since there are no common factors between 1 and 81, except for 1, the fraction is considered to be simplified as much as possible in this case. Therefore, the final process is done and we have our answer.

Key Concepts

Exponent Rules

Negative exponents can be tricky, especially when they involve fractions. In essence, a negative exponent means taking the reciprocal of the base and applying the corresponding positive exponent. In simpler terms, if we have a number raised to a negative power, we flip it to get a fraction.

For example, if we have \((-27)^{-\frac{4}{3}}\),the negative exponent tells us to find the reciprocal of the whole number. So, \(-27^{-\frac{4}{3}} = 1 / 27^{\frac{4}{3}}\).

When dealing with negative bases, remember that a double negative results in a positive number. This is why combining the base of \(-27\) and the negative exponent produces \(1 / 27^{\frac{4}{3}}\).The exponent can then be managed using fractional powers by taking cube roots first, and subsequently raising to the required power.

Cube Roots

Cube roots are a type of root calculation that deals with finding a value which, when multiplied by itself three times, equals the original number. In our example, the expression \(27^{\frac{1}{3}}\)asks us to find the cube root of 27.

To evaluate this:

• Identify a number that, when cubed, gives 27. This number is 3, as \(3 \times 3 \times 3 = 27\).

This step is crucial when dealing with fractional exponents because it breaks down complicated expressions into simpler pieces. After finding the cube root, we then use the result in further calculations, such as raising it to another power as indicated by the remaining part of the exponent.

Fraction Simplification

Once we've applied any necessary operations from exponent rules and cube roots, we often end up with a fraction. Simplifying this fraction is the final task to ensure our answer is in its cleanest form.

Consider our fraction \(\frac{1}{81}\).To simplify:

• Check for any common factors in the numerator and denominator. With \(1\) and \(81\),the only common factor is 1.

This means our fraction is already simplified. Sometimes, you might work with larger numbers where simplification involves reducing by common factors, but in this case, \(\frac{1}{81}\)is as simple as it can be. This final step is key in arriving at the clearest possible result.