Examples with solutions for Power of a Quotient Rule for Exponents: Solving the problem

Exercise #1

Solve the following expression:

406736490=? \frac{4^0\cdot6^7}{36^4\cdot9^0}=\text{?}

Video Solution

Step-by-Step Solution

When raising any number to the power of 0 it results in the value 1, mathematically:

X0=1 X^0=1

Apply this to both the numerator and denominator of the fraction in the problem:

406736490=1673641=67364 \frac{4^0\cdot6^7}{36^4\cdot9^0}=\frac{1\cdot6^7}{36^4\cdot1}=\frac{6^7}{36^4}

Note that -36 is a power of the number 6:

36=62 36=6^2

Apply this to the denominator to obtain expressions with identical bases in both the numerator and denominator:

67364=67(62)4 \frac{6^7}{36^4}=\frac{6^7}{(6^2)^4}

Recall the power rule for power of a power in order to simplify the expression in the denominator:

(am)n=amn (a^m)^n=a^{m\cdot n}

Recall the power rule for division between terms with identical bases:

aman=amn \frac{a^m}{a^n}=a^{m-n}

Apply these two rules to the expression that we obtained above:

67(62)4=67624=6768=678=61 \frac{6^7}{(6^2)^4}=\frac{6^7}{6^{2\cdot4}}=\frac{6^7}{6^8}=6^{7-8}=6^{-1}

In the first stage we applied the power of a power rule and proceeded to simplify the expression in the exponent of the denominator term. In the next stage we applied the second power rule - The division rule for terms with identical bases, and again simplified the expression in the resulting exponent.

Finally we'll use the power rule for negative exponents:

an=1an a^{-n}=\frac{1}{a^n}

We'll apply it to the expression that we obtained:

61=16 6^{-1}=\frac{1}{6}

Let's summarize the various steps of our solution:

406736490=16 \frac{4^0\cdot6^7}{36^4\cdot9^0}=\frac{1}{6}

Therefore the correct answer is A.

Answer

16 \frac{1}{6}

Exercise #2

Solve the following:

35xy77xy8x5y= \frac{35x\cdot y^7}{7xy}\cdot\frac{8x}{5y}=

Video Solution

Step-by-Step Solution

To solve this problem, follow these steps:

Step 1: Simplify the first fraction:

The first expression is 35xy77xy\frac{35x \cdot y^7}{7xy}.

  • Cancel the common factor of 77: 35÷7=535 \div 7 = 5.

  • This simplifies to 5xy7xy\frac{5x \cdot y^7}{x \cdot y}.

  • Cancel the common factor of xx: x/xx/x cancels to 11.

  • Cancel part of the yy terms: y7/y=y71=y6y^7/y = y^{7-1} = y^6.

  • The result is 5y65y^6.

Step 2: Simplify the second fraction:

The second expression is 8x5y\frac{8x}{5y}.

  • No common factors in the numerator and denominator, so it remains 8x5y \frac{8x}{5y} .

Step 3: Multiply these simplified results:

Now, multiply the results from Step 1 and Step 2: 5y68x5y5y^6 \cdot \frac{8x}{5y}.

  • The factor of 55 in 5y65y^6 and 8x5y\frac{8x}{5y} cancels: 5/5=15/5 = 1.

  • This results in y68xyy^6 \cdot \frac{8x}{y}.

  • Cancel part of the yy terms: y6/y=y61=y5y^6/y = y^{6-1} = y^5.

Thus, the simplified expression is 8xy58xy^5.

Therefore, the solution to the problem is 8xy5 \mathbf{8xy^5} .

Answer

8xy5 8xy^5

Exercise #3

Solve:

8x7y32042x5y2= \frac{8x^7y^3}{20}\cdot\frac{4}{2x^5y^2}=

Video Solution

Answer

4x2y5 \frac{4x^2y}{5}