Problem 40 - Contemporary Precalculus

Question

Simplify the expression without using a calculator. $$\sqrt{54 m^{-6} n^{3}}$$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\sqrt{54 m^{-6} n^{3}}$$ without using a calculator. Answer: The simplified expression is $$3\sqrt{6}m^{-3}n^{\frac{3}{2}}$$.

Step 1: Rewrite the expression with rational exponents

Instead of using a square root symbol, rewrite the expression using a rational exponent of 1/2: $$\sqrt{54 m^{-6} n^{3}} = (54 m^{-6} n^{3})^{\frac{1}{2}}$$

Step 2: Distribute the exponent

Apply the power rule of exponents to each term within the parentheses. Power rule: \((ab)^c = a^c b^c\) Remember that the original exponent should be multiplied by the outer exponent: $$ (54 m^{-6} n^{3})^{\frac{1}{2}} = 54^{\frac{1}{2}}m^{-6\cdot\frac{1}{2}}n^{3\cdot\frac{1}{2}}$$

Step 3: Simplify the expression

Simplify the exponents and, if possible, the numerical term (in this case, 54 has a perfect square factor): $$54^{\frac{1}{2}} = \sqrt{54} = \sqrt{9\cdot6} = \sqrt{9}\sqrt{6}=3\sqrt{6}$$ $$m^{-6\cdot\frac{1}{2}} = m^{-3}$$ $$n^{3\cdot\frac{1}{2}} = n^{\frac{3}{2}}$$ Combine all the simplified terms: $$3\sqrt{6}m^{-3}n^{\frac{3}{2}}$$ The simplified expression without using a calculator is: $$3\sqrt{6}m^{-3}n^{\frac{3}{2}}$$

Key Concepts

Rational Exponents

Working with radical expressions can often be simplified through the use of rational exponents. But what exactly are rational exponents? Simply put, a rational exponent is an exponent that is a fraction. For example, the square root of a number is the same as raising that number to the power of \frac{1}{2}. When you see an expression like \( \sqrt{x} \), you can rewrite it as \( x^{\frac{1}{2}} \).

This is tremendously helpful because it allows us to apply the rules of exponents that we're familiar with to expressions under a root. In the exercise given, \( \sqrt{54 m^{-6} n^{3}} \) becomes \( (54 m^{-6} n^{3})^{\frac{1}{2}} \), making it easier to simplify the expression using the power rule of exponents. It's important to understand this concept because it streamlines the process of simplifying radical expressions, especially as they become more complex.

Power Rule of Exponents

The power rule of exponents is a cornerstone concept in algebra that simplifies the process of raising powers to powers. If you have an expression like \( (a^b)^c \) you can multiply the exponents to simplify the expression to \( a^{b\cdot c} \). Applying this rule to our main exercise, we see how the exponent \( \frac{1}{2} \) is distributed to each factor inside the parentheses.

So, when you review the step by step solution, take note that the power rule is applied as follows: \( (54 m^{-6} n^{3})^{\frac{1}{2}} = 54^{\frac{1}{2}}m^{-6\cdot \frac{1}{2}}n^{3\cdot \frac{1}{2}} \). By understanding the power rule of exponents, you can tackle more complex expressions with confidence, knowing that you can break them down and simplify each part in an orderly manner.

Simplifying Square Roots

Simplifying square roots can sometimes feel like a magic trick, turning something complicated into a more manageable form. The trick is to look for perfect square factors within the number. A perfect square is a number that has an integer square root, such as 4, 9, 16, and so on. In our original exercise, we tackle \( 54^\frac{1}{2} \) by breaking down 54 into \( 9 \times 6 \) because 9 is a perfect square.

Once you've identified the perfect square, you can take the square root of that part separately. So, \( \sqrt{54} = \sqrt{9\cdot6} = \sqrt{9}\sqrt{6}=3\sqrt{6} \). This simplification method also applies to variables with even exponents. Always remember: find the perfect squares, simplify them, and multiply by the remaining square root, if any. This approach makes it much easier to work with and understand square roots without relying on a calculator.