Problem 43 - Contemporary Precalculus

Question

Simplify the expression without using a calculator. $$\frac{\sqrt[3]{a^{5} b^{4} c^{3}}}{\sqrt[3]{a^{-1} b^{2} c^{6}}}$$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\frac{\sqrt[3]{a^{5} b^{4} c^{3}}}{\sqrt[3]{a^{-1} b^{2} c^{6}}}$$ without using a calculator. Answer: $$\frac{a^2b^{2/3}}{c}$$

Step 1: Combine the cube roots

Since the expression is a fraction with cube roots in both numerator and denominator, we can combine the cube roots using the rule $$\frac{\sqrt[3]{A}}{\sqrt[3]{B}} = \sqrt[3]{\frac{A}{B}}$$. Applying this rule, we get: $$\frac{\sqrt[3]{a^{5} b^{4} c^{3}}}{\sqrt[3]{a^{-1} b^{2} c^{6}}} = \sqrt[3]{\frac{a^{5} b^{4} c^{3}}{a^{-1} b^{2} c^{6}}}$$

Step 2: Simplify within the cube root

Now, we need to simplify the expression within the cube root. We can do this by dividing each term with the same base in the numerator by the corresponding term in the denominator. Recall that when we divide terms with the same base, we subtract their exponents. So, we have: $$\sqrt[3]{\frac{a^{5} b^{4} c^{3}}{a^{-1} b^{2} c^{6}}} = \sqrt[3]{a^{5-(-1)}b^{4-2}c^{3-6}}$$ Simplify the exponents: $$\sqrt[3]{a^{6}b^{2}c^{-3}}$$

Step 3: Move terms with negative exponents

Now, we simplify the cube root expression by moving the term with a negative exponent to the denominator. Recall that a term with a negative exponent can be rewritten with a positive exponent in the reciprocal position: $$c^{-3} = \frac{1}{c^3}$$ So, the simplified expression is: $$\frac{\sqrt[3]{a^{6}b^{2}}}{\sqrt[3]{c^{3}}}$$

Step 4: Simplify the cube roots

Finally, we can simplify the cube root expressions. The cube roots simplify as follows: $$\sqrt[3]{a^{6}} = a^{6/3} = a^2$$ $$\sqrt[3]{b^{2}} = b^{2/3}$$ $$\sqrt[3]{c^{3}} = c^{3/3} = c$$ Thus, the simplified expression is: $$\frac{a^2b^{2/3}}{c}$$

Key Concepts

Cube Roots

A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, if you have \(x^3 = y\), then \(x = \sqrt[3]{y}\). Cube roots are useful for simplifying expressions where terms are raised to powers that are multiples of three.
To simplify a quotient of cube roots like \(\frac{\sqrt[3]{A}}{\sqrt[3]{B}}\), you can use the property \(\sqrt[3]{\frac{A}{B}}\). This allows both the numerator and the denominator to be combined under a single cube root. Using this method can make complex expressions simpler and more manageable.

Exponents

Exponents are numbers that denote repeated multiplication of a base number. If you see \(a^n\), it means you have the base 'a' multiplied by itself 'n' times.
For example, in the expression \(a^5\), 'a' is the base and 5 is the exponent, which tells you to multiply 'a' by itself 5 times. Exponents follow specific rules, such as when multiplying like bases, you add their exponents, and when dividing like bases, you subtract their exponents. These rules are essential for simplifying algebraic expressions.

Negative Exponents

Negative exponents are a way to represent division or reciprocal. When you see an expression like \(a^{-n}\), it denotes \(\frac{1}{a^n}\).
For example, \(a^{-1} = \frac{1}{a}\). This is handy for transforming terms with negative exponents to positive exponents by moving them to the opposite side of a fraction. This is evident from our original problem: moving \(c^{-3}\) to the denominator as \(\frac{1}{c^3}\). Remember, exchanging negative exponents in the numerator to positive ones in the denominator, or vice versa, can simplify expressions.

Fractional Exponents

Fractional exponents provide a way to express roots, like cube roots, in a different form. An expression like \(a^{\frac{m}{n}}\) can be converted to the nth root of the base raised to the mth power, \(\sqrt[n]{a^m}\).
As in our solution, after simplifying the cube root, \(\sqrt[3]{a^6}\) becomes \(a^{6/3} = a^2\). Likewise, \(b^{2/3}\) indicates the cube root of \(b^2\). Fractional exponents simplify operations and calculations involving roots, making the expressions easier to manipulate and understand.