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Chapter 9: Problem 133

Simplify. $$ \sqrt{\frac{100 x^{5}}{36 x^{3}}} $$

Short Answer

Expert verified
\( \frac{5x}{3} \)

Step by step solution

01

- Simplify the fraction inside the square root

The given expression is \[ \sqrt{\frac{100 x^{5}}{36 x^{3}}}\] Simplify the fraction inside the square root by dividing the coefficients and subtracting the exponents of like bases. \[ \frac{100 x^{5}}{36 x^{3}} = \frac{100}{36} \cdot \frac{x^{5}}{x^{3}} = \frac{25}{9} \cdot x^{2} \]
02

- Apply the square root to the simplified fraction

Next, apply the square root to both the numerator and denominator separately: \[ \sqrt{\frac{25 x^{2}}{9}} = \frac{\sqrt{25 x^{2}}}{\sqrt{9}} \]
03

- Simplify the square root

Now, simplify the square roots of the numerator and the denominator: \[ \sqrt{25 x^{2}} = 5x \] and \[ \sqrt{9} = 3 \] Therefore, \[ \frac{\sqrt{25 x^{2}}}{\sqrt{9}} = \frac{5x}{3} \]
04

- Final simplified result

Combine all the simplified parts to get the final answer: \[ \sqrt{\frac{100 x^{5}}{36 x^{3}}} = \frac{5x}{3} \]

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Key Concepts

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

Fraction Simplification
To simplify a fraction, we can break it down into simpler parts that are easier to work with. In the expression \( \frac{100 x^{5}}{36 x^{3}} \), this involves two key steps: simplifying the coefficients and the variables.
First, we handle the coefficients (the numerical parts). We divide 100 by 36 to get \(\frac{25}{9}\).
Next, simplify the variables with exponents. We use the rule of subtracting exponents: \(x^{5} \div x^{3} = x^{5-3} = x^{2}\).
Combining these gives \( \frac{25 x^{2}}{9} \). Breaking down a complex fraction into simpler components like this makes it easier to manage.
Exponent Rules
Understanding exponent rules is key to solving expressions like \(\frac{100 x^{5}}{36 x^{3}}\). These rules guide us in simplifying variables with exponents.
Here are some important exponent rules to remember:
  • Product of Powers Rule: When multiplying like bases, add the exponents: \( a^m \times a^n = a^{m+n} \).
  • Quotient of Powers Rule: When dividing like bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n} \). This is what we used in \(\frac{x^{5}}{x^{3}} = x^{2} \).
  • Power of a Power Rule: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m \times n} \).

These basic rules simplify expressions and make them more manageable.
Square Root Properties
Square roots simplify expressions by finding the original number that, when multiplied by itself, gives the number under the root. We use these properties often in algebra.
For example, in \(\frac{\text{sqrt}{25 x^{2}}}{\text{sqrt}{9}}\):
  • Simplifying the numerator: \(\text{sqrt}{25 x^{2}} = \text{sqrt}{25} \times \text{sqrt}{x^{2}} = 5 \times x = 5x \).
  • Simplifying the denominator: \(\text{sqrt}{9} = 3 \).
    Combining these results: \(\frac{5x}{3} \).
    Overall, square root properties help break down complex expressions into simpler parts that are easier to solve.

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