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Chapter 11: Problem 50

Write each expression in simplest radical form. If \(a\) radical appears in the denominator, rationalize the denominator. \(\sqrt{98 x^{6} y^{-7}}\)

Short Answer

Expert verified
The simplest radical form is \( \frac{7x^3 \sqrt{2y}}{y^4} \).

Step by step solution

01

Simplify the Radicand

Start by simplifying the expression inside the square root: \( \sqrt{98x^6y^{-7}} \). Break down the components: 98 can be factored into \( 2 \times 49 \) and 49 is a perfect square (\( 7^2 \)). The expression becomes \( \sqrt{2 \times 7^2 \times (x^6) \times (y^{-7})} \).
02

Simplify the Squared Terms

Extract the square terms from the square root. \( 7^2 \) simplifies to 7 outside the root, and \( x^6 \) becomes \( x^3 \) because \( x^6 = (x^3)^2 \). The expression is now \( 7x^3\sqrt{2y^{-7}} \).
03

Simplify the Negative Exponent

Convert the negative exponent by using the property \( y^{-7} = \frac{1}{y^7} \). The expression inside the square root becomes \( \sqrt{\frac{2}{y^7}} = \frac{\sqrt{2}}{\sqrt{y^7}} \).
04

Rationalize the Denominator

To simplify \( \frac{\sqrt{2}}{\sqrt{y^7}} \), rationalize the denominator. Rewrite \( \sqrt{y^7} \) as \( y^3 \sqrt{y} \), so \( \frac{\sqrt{2}}{y^3 \sqrt{y}} \). Multiply numerator and denominator by \( \sqrt{y} \) to get \( \frac{\sqrt{2y}}{y^4} \).
05

Combine the Terms

Finally, combine the terms outside and inside the radical. The result is \( \frac{7x^3 \sqrt{2y}}{y^4} \). This is the expression in its simplest radical form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rationalize the Denominator
Rationalizing the denominator involves eliminating any radicals from the bottom of a fraction. This process is crucial because it is often seen as a standard way to present a fraction, making it easier to understand and work with. To rationalize a denominator like \( \frac{\sqrt{2}}{y^3 \sqrt{y}} \), which has a radical in it, you multiply both the numerator and the denominator by the same radical that will help remove it. In this case, multiplying by \( \sqrt{y} \) transforms the expression to \( \frac{\sqrt{2y}}{y^4} \).
  • Multiply both the top and bottom by the same square root.
  • This should result in a rational number in the denominator.
This technique ensures that when expressions are simplified, they are both mathematically accurate and conventionally acceptable.
Negative Exponents
A negative exponent indicates taking the reciprocal of a base raised to a positive exponent. This can be confusing initially, but with some practice becomes an intuitive part of algebra. For example, the expression \( y^{-7} \) is equivalent to \( \frac{1}{y^7} \). This is because any base with a negative exponent can be rewritten as a fraction:
  • Positive Exponents: \( a^3 \)
  • Negative Exponents: \( a^{-3} = \frac{1}{a^3} \)
Converting negative exponents is an essential step in simplifying expressions, as it paves the way for further simplification, like rationalization.
Perfect Squares
Perfect squares are numbers or expressions that can be written as the square of another number or expression. Recognizing perfect squares can be a huge time saver when simplifying radical expressions.Understanding them makes it straightforward to determine what can be pulled out of a radical. For instance, in the expression \( 49 \), because \( 49 = 7^2 \), we can directly take 7 out of the radical. Similarly, if you have \( x^6 = (x^3)^2 \), you can simplify this to \( x^3 \) outside the square root as well.
  • A perfect square provides a whole number when taken as a square root.
  • Identifying them speeds up simplification processes.
Using perfect squares effectively can reduce complex expressions into much simpler forms, allowing for easier manipulation and understanding.
Square Root Simplification
To simplify a square root, break down the number or expression inside into its prime factors or identifiable components. This is done to separate the perfect squares from the rest.Take \( \sqrt{98x^6y^{-7}} \), for instance. By expressing 98 as \( 2 \times 49 \), and recognizing \( 49 \) as a perfect square, you simplify it to \( 7 \) outside the radical. Similarly, split \( x^6 \) into \( (x^3)^2 \) to get \( x^3 \) out.
  • Identify the factors within the radical.
  • Extract perfect squares.
  • Reconstruct the expression to its simplest form.
Simplifying square roots makes calculations simpler and is a valuable skill for solving algebraic expressions.

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Most popular questions from this chapter

Solve the given problems. When studying the orbits of earth satellites, the expression \(\left(\frac{G M}{4 \pi^{2}}\right)^{1 / 3} T^{2 / 3}\) arises. Express it in simplest rationalized radical form.

In Exercises \(5-52,\) express each of the given expressions in simplest form with only positive exponents. $$3 a^{-2}+\left(3 a^{-2}\right)^{4}$$

In Exercises \(5-52,\) express each of the given expressions in simplest form with only positive exponents. $$2\left(5 a n^{-2}\right)^{-1}$$

In Exercises \(5-52,\) express each of the given expressions in simplest form with only positive exponents. $$(-4)^{0}$$

Solve the given problems. Are there any values of \(x\) or \(y\) such that \(\sqrt{x}+\sqrt{y}=\sqrt{x+y} ?\)
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