Problem 94 - Algebra for College Students

Question

Definition \(5.7\) states that $$ b^{\frac{m}{n}}=\sqrt[n]{b^{m}}=(\sqrt[n]{b})^{m} $$ Use your calculator to verify each of the following. (a) \(\sqrt[3]{27^{2}}=(\sqrt[3]{27})^{2}\) (b) \(\sqrt[3]{8^{5}}=(\sqrt[3]{8})^{5}\) (c) \(\sqrt[4]{16^{3}}=(\sqrt[4]{16})^{3}\) (d) \(\sqrt[3]{16^{2}}=(\sqrt[3]{16})^{2}\) (e) \(\sqrt[5]{9^{4}}=(\sqrt[5]{9})^{4}\) (f) \(\sqrt[3]{12^{4}}=(\sqrt[3]{12})^{4}\)

Step-by-Step Solution

Answer

(a), (b), (c), and (f) are equal; (d) and (e) are not equal.

Step 1: Calculate Cube Root and Power for (a)

For the expression \( \sqrt[3]{27^2} \), first calculate the cube root of 27. This is 3 because \( 3^3 = 27 \). Then, calculate \( 3^2 = 9 \). Simultaneously compute \( (\sqrt[3]{27})^2 = 3^2 = 9 \). Since both expressions yield 9, they are equal.

Step 2: Calculate Cube Root and Power for (b)

For \( \sqrt[3]{8^5} \), calculate the cube root of 8, which is 2 because \( 2^3 = 8 \). Then, compute \( 2^5 \). \( 2^5 = 32 \). Compute \( \sqrt[3]{8^5} = (64)^{1/3} \) which also equals 32.

Step 3: Calculate Fourth Root and Power for (c)

For \( \sqrt[4]{16^3} \), calculate the fourth root of 16, which is 2 because \( 2^4 = 16 \). Then calculate \( 2^3 = 8 \). Compute \((\sqrt[4]{16})^3 = 8 \) as well. Both expressions are equal.

Step 4: Calculate Cube Root and Power for (d)

For \( \sqrt[3]{16^2} \), calculate the cube root of 16 approximately, which is near \( 2.5198 \). Compute \( (2.5198)^2 \), which approximates 6.35. Calculate it directly to match: \( \sqrt[3]{256} \approx 6.349 \), close enough to verify equality when rounded.

Step 5: Calculate Fifth Root and Power for (e)

For \( \sqrt[5]{9^4} \), calculate the fifth root of 9 which is approximately \( 1.5518 \). Then calculate \( (1.5518)^4 \). Directly compute \( \sqrt[5]{9^4} = (6561)^{1/5} \), which equals approximately \( 6.56 \), not matching the first value indicating inequality.

Step 6: Calculate Cube Root and Power for (f)

For \( \sqrt[3]{12^4} \), calculate the cube root of 12, which is approximately \( 2.289 \). Then calculate \( (2.289)^4 \) which yields \( 27.376 \). Calculate \( \sqrt[3]{20736} \approx 27.31 \), demonstrating very close values, showing equality at precision limit.

Key Concepts

Rational Exponents

Rational exponents are a way of expressing powers and roots together within the same expression. Instead of showing the operation as a traditional root or power, you can use fractional exponents to simplify calculations. The general format for a rational exponent is \( b^{\frac{m}{n}} \). The fraction \( \frac{m}{n} \) tells us to take the nth root of \( b \) and then raise the result to the mth power. For example, if you encounter \( 27^{2/3} \), it means: take the cube root of 27, which gives you 3, and then square it to get 9.

This approach can simplify calculations and make it easier to manipulate expressions involving both powers and roots. Whether you're dealing with cube roots, square roots, or even more complex fractions, understanding how to use rational exponents is crucial in algebra.

Cube Root

A cube root of a number \( x \) is any number \( y \) such that \( y^3 = x \). To find a cube root, you are essentially answering the question, "What number, when multiplied by itself three times, produces \( x \)?" For example, to find the cube root of 27, consider what number cubed equals 27. The answer is 3 because \( 3^3 = 27 \).

• Cube roots are often expressed with the radical symbol \( \sqrt[3]{x} \).
• In the context of rational exponents, \( x^{1/3} \) is equivalent to the cube root of \( x \).

Cubes and cube roots play an important role in many areas of mathematics, including geometry and solving polynomial equations. By mastering the calculations related to cube roots, you'll better handle and simplify expressions that might initially seem complex.

Fourth Root

The fourth root of a number \( x \) is a value \( y \) that satisfies \( y^4 = x \). To find a fourth root, you determine which number in the fourth power results in \( x \). For example, the fourth root of 16 is 2 because \( 2^4 = 16 \).

• Fourth roots can be written as \( \sqrt[4]{x} \).
• In rational exponent form, this is expressed as \( x^{1/4} \).

When you work with higher roots, understanding these expressions becomes crucial, especially in simplifying higher degree polynomial equations and in calculus. Fourth roots, much like square and cube roots, are pivotal when conducting more advanced algebraic manipulations.

Fifth Root

The fifth root of a number \( x \) is any value \( y \) such that \( y^5 = x \). This means you are looking for a number which multiplied by itself five times equals \( x \). As an example, consider the fifth root of 32: the answer is 2, as \( 2^5 = 32 \).

• Expressed with a radical, this is \( \sqrt[5]{x} \).
• Rational exponents allow us to note the fifth root as \( x^{1/5} \).

Fifth roots are used in advanced mathematical problems including certain forms of polynomial roots and in higher degree equations. Becoming familiar with calculating and using fifth roots is important for solving complex real-world mathematical challenges.