Problem 111 - College Algebra

Question

Simplify each expression. Assume that all variables represent positive numbers. $$ \left(49 x^{-2} y^{4}\right)^{-\frac{1}{2}}\left(x y^{\frac{1}{2}}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{x^{2}}{7y^{1.5}} \).

Step 1: Simplify Exponent

First, simplify the exponent \( -\frac{1}{2}\) in the term \( (49x^{-2}y^{4})^{-\frac{1}{2}} \). This can be done by raising each part of the term inside the parentheses to the power of \( -\frac{1}{2} \). So, \( (49x^{-2}y^{4})^{-\frac{1}{2}} = \left(49^{-\frac{1}{2}}\right)\left(x^{-(-2 \cdot \frac{1}{2})}\right)\left(y^{4 \cdot -\frac{1}{2}}\right) = \left(\sqrt{\frac{1}{49}}\right)x^{1}y^{-2} \).

Step 2: Perform Multiplication

The next step would be to multiply the result from step 1 by the second term \(xy^{\frac{1}{2}}\). So, \( \left(\sqrt{\frac{1}{49}}\right)x^{1}y^{-2} \times xy^{\frac{1}{2}} = \frac{1}{7}x^{2}y^{-1.5} \).

Step 3: Simplify your Expression

Now, we'll convert negative and fractional exponents to positive as follows: \( \frac{1}{7}x^{2}y^{-1.5} = \frac{x^{2}}{7y^{1.5}} \). Thus, the simplified expression is \( \frac{x^{2}}{7y^{1.5}} \).

Key Concepts

Negative Exponents

Negative exponents can be a bit tricky, but they become much easier once you understand them. A negative exponent means that instead of multiplying the base by itself, you take the reciprocal of the base raised to the positive of that exponent. For example, if you have an expression like

• \( x^{-2} \), it translates to \( \frac{1}{x^2} \).

This means you're essentially "flipping" the number. Any non-zero number raised to a negative exponent will give you the reciprocal of that number raised to the corresponding positive exponent.
It's a useful rule when trying to simplify expressions, especially in algebra. By converting negative exponents into fractions, you can often combine and simplify parts of expressions in ways that are not immediately obvious.
In our problem, the negative exponent \(-\frac{1}{2}\) was part of the expression \( 49x^{-2}y^4 \), which we simplified by taking reciprocals and changing the signs of the exponents. Consequently, \( 49^{-\frac{1}{2}} \) becomes \( \frac{1}{\sqrt{49}} \) and \( x^{2} \) found its way to the numerator as \( x\).

Fractional Exponents

Fractional exponents are another crucial concept in algebra. They are related to roots or radicals. A fractional exponent is actually a way to represent roots, where the numerator indicates the power and the denominator indicates the root. An expression like

• \( x^{\frac{1}{2}} \) means the square root of \( x \), or \( \sqrt{x} \).

This is an efficient and concise way to express roots in mathematical expressions.
In the step-by-step solution, you saw this in action. The expression \( \left( 49^{ -\frac{1}{2} } \right) \) was converted into \( \sqrt{ \frac{1}{49} } \). The \( -\frac{1}{2} \) made it both a root and a reciprocal, emphasizing the importance of understanding how fractional exponents work in tandem with negative exponents.
In algebra, you'll frequently encounter these types of exponents, so grasping the concept helps in handling a variety of mathematical expressions.

Algebraic Expressions

Algebraic expressions can include variables, coefficients, and exponents and they are the bread and butter of algebra. Algebraic expressions are combinations of numbers and letters linked by mathematical operations like addition, subtraction, multiplication, etc. Simplifying them is essential, as understanding this makes tackling more complex equations manageable.
This involves applying rules like those for negative and fractional exponents, as seen in our original exercise.
The given expression combined terms with different exponents:

• \( x^{-2} \), \( y^{4} \), and \( y^{\frac{1}{2}} \).

Your job was to apply the rules for exponents, identifying like terms to simplify the expression fully.
In the final expression, you saw how everything came together, illustrating the beauty of algebraic manipulation.
Understanding how to simplify algebraic expressions equips you to solve more complex problems in your math journey.