Problem 22 - Algebra 2

Question

Simplify. $$ \sqrt[3]{-0.064} $$

Step-by-Step Solution

Answer

The cube root of \(-0.064\) is \(-\frac{4}{5}\).

Step 1: Identify the Cube Root Operation

The given expression is \( \sqrt[3]{-0.064} \), which means we need to find a number that, when raised to the power of 3, gives \(-0.064\). This is called finding the cube root.

Step 2: Convert to a Fraction

Convert \(-0.064\) into a fraction form. \(-0.064 = -\frac{64}{1000}\), which simplifies to \(-\frac{64}{1000} = -\frac{16}{250}\) by dividing numerator and denominator by 4. Further simplifying gives \(-\frac{4}{62.5}\). Finally, it reduces to \(-\frac{1}{2.5}\).

Step 3: Consider Simple Fractions

Recognize \( -0.064 \) is \(-\left(\frac{4}{5}\right)^3\). This gives us a potential candidate for cube root because \(\frac{4}{5}\) is both a simple and common fraction, which in this context, can help simplify the cube root evaluation.

Step 4: Calculate the Cube Root

Finding the cube root of \(-\left(\frac{4}{5}\right)^3\), use the property \( \sqrt[3]{x^3} = x \), which gives us \( -\frac{4}{5} \) as the cube root of \(-\left(\frac{4}{5}\right)^3\).

Step 5: Verifying the Result

Check that \(-\frac{4}{5}^3 = -0.064\). Raising \(-\frac{4}{5}\) to the power of 3 gives \(-\left(\frac{4}{5}\right)^3 = -\frac{64}{125} = -0.064\), confirming that \(-\frac{4}{5}\) is indeed the correct cube root of \(-0.064\).

Key Concepts

Simplifying Fractions

Simplifying fractions means finding an equivalent fraction in its simplest form. This is a fraction where the numerator (top number) and the denominator (bottom number) are as small as possible but still have the same value as the original fraction. To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD). Using the example from the exercise, consider the fraction \(-\frac{64}{1000}\). Both 64 and 1000 can be divided evenly by 4. When you divide both by 4, you get \(-\frac{16}{250}\). By continuing this process, you'll eventually simplify it to \(-\frac{4}{62.5}\) and finally to \(-\frac{1}{2.5}\).

• Identify if both numbers can be divided by the same number.
• Continue dividing until they can't be reduced further.
• The fraction is simplified when no further division is possible without leaving a remainder.

By simplifying fractions, calculations become easier and more straightforward.

Fraction to Decimal Conversion

Converting a fraction to its decimal form involves dividing the numerator by the denominator. This can be done using long division or a calculator for simplicity. In our exercise, the fraction \(-\frac{64}{1000}\) can be directly converted into the decimal \-0.064\. Here's why:

• The denominator, 1000, is a power of 10, which simplifies division.
• Since 64 is divided by 1000, move the decimal three places to the left.
• This results in \-0.064\ (negative due to the negative sign in the fraction).

In simpler terms, anytime you have a fraction where the denominator is a power of 10, you can quickly turn it into a decimal by shifting the decimal point in the numerator left by the number of zeros in the denominator.

Exponents

Exponents are a way to express repeated multiplication of the same number. In the context of cube roots and cube calculations, an exponent of 3 indicates that a number is multiplied by itself twice more \((x^3 = x \times x \times x)\). Understanding this concept is crucial for finding cube roots as well. When you encounter \(-\left(\frac{4}{5}\right)^3\), it means multiplying \-\frac{4}{5}\ two times more by itself. This is crucial when working with cube roots because:

• The cube root of a number is what number times itself three times gives you the original number.
• In the exercise, the cube root of \-\left(\frac{4}{5}\right)^3\, simplifies back to \-\frac{4}{5}\.
• This process helps verify our calculations and ensure they're correct.

By understanding and manipulating exponents, you can solve a variety of mathematical problems, including those involving roots and powers.