Writing Equivalent Algebraic Expressions For Multiplication and Division

Writing Equivalent Algebraic Expressions For Multiplication and Division exercise

• Explain how to use the quotient of powers property for like bases.

  Hints

  Keep the following powers in mind

  • $a^1=a$
  • $a^0=1$ for $a\neq 0$

  Keep the order of subtraction in mind:

  $\frac{a^4}{a^2}=\frac{a\cdot a\cdot \not a\cdot \not a}{\not a\cdot \not a}=\frac{a^2}1=a^2$

  Remember the fraction notation:

  $\frac{\text{numerator}}{\text{denominator}}$

  Remember that negative just means opposite. So if an exponent is positive we already know that means how many times we multiply the number. Therefore, a negative exponent means how many times to divide by the number.

  For example $c^{-2} = \frac{1}{c \cdot c}=\frac{1}{c^2}$

  Solution

  First, let's have a look at the left stamps with the letter $a$. The corresponding expression is

  $\frac{a^3}{a^3}=\frac{\not a\cdot \not a\cdot \not a}{\not a\cdot \not a\cdot \not a}=\frac 11=1$

  How can we generalize this? You see the like basis $a$. You can cancel out as many factors as the highest in the numerator as well as the denominator:

  $\frac{a^3}{a^3}=a^{3-3}=a^0=1$

  Wow, that's it. You just have to subtract the exponent in the denominator from this one in the numerator. The base stays the same.

  It is helpful to memorize that: $a^n$ is called a power, where $a$ is the base and $n$ the exponent.

  Let's try it one more time:

  $\frac{b^2}{b}=\frac{b\cdot \not b}{\not b}=b^1=b$

  Does this also work with the property above?

  $\frac{b^2}{b}=b^{2-1}=b^1=b$

  Yes! This property is called the quotient of powers property and works when the bases are the same

  $\frac{a^m}{a^n}=a^{m-n}$

  Just keep in mind that order matters with subtraction.

  Last we check if this property also works if the exponent in the denominator is greater than this one in the numerator:

  $\frac{c^4}{c^5}=\frac{\not c\cdot \not c\cdot \not c\cdot \not c}{\not c\cdot \not c\cdot \not c\cdot \not c\cdot \not c}=\frac 1{c}$

  Now we use the quotient of powers property

  $\frac{c^4}{c^5}=c^{4-5}=c^{-1}=\frac1c$ $~~~~~$✓

• Simplify the expressions using the quotient of powers property.

  Hints

  Here you see the quotient of powers property for like bases.

  Be careful to subtract the exponent in the denominator from the one in the numerator and not the other way round.

  Here is an example of the quotient of powers property.

  Solution

  Let's practice the quotient of powers property with a few examples:

  $\mathbf{\frac{a^2}a}$

  We just have to subtract the exponents: this one in the denominator from this one in the numerator to get

  $\frac{a^2}{a}=a^{2-1}=a^1=a$

  $\mathbf{27a^2b^3\div(9ab^2)=\frac{27a^2b^3}{9ab^2}}$

  Remember: The fraction bar is just another way to write down a division.

  We first rearrange this term to get only fractions of like terms:

  $\frac{27a^2b^3}{9ab^2}=\frac{27}9\cdot\frac{a^2}{a}\cdot\frac{b^3}{b^2}$

  Now we simplify each fraction step by step

  • $\frac{27}9=3$
  • $\frac{a^2}{a}=a^{2-1}=a^1=a$
  • $\frac{b^3}{b^2}=b^{3-2}=b^1=b$

  Thus we get

  $\frac{27a^2b^3}{9ab^2}=3ab$

  $\mathbf{18x^5\div\frac{6x^3y^2}{y^3}=\frac{18x^5}{1}\cdot\frac{y^3}{6x^3y^2}}$

  Once again we rearrange the terms

  $\frac{18x^5}{1}\cdot\frac{y^3}{6x^3y^2}=\frac{18}{6}\cdot\frac{x^5}{x^3}\cdot\frac{y^3}{y^2}$

  Now we are in the position to use the quotient of powers property for like basis

  • $\frac{18}6=3$
  • $\frac{x^5}{x^3}=x^{5-3}=x^2$
  • $\frac{y^3}{y^2}=y^{3-2}=y^1=y$

  Thus we get

  $18x^5\div\frac{6x^3y^2}{y^3}=3x^2y$

• Determine the value of each expression.

  Hints

  First, use the quotient of powers property for like basis.

  Remember the exponents mean multiplication by itself that many times. For example, $6^2 = 6 \cdot 6 = 36$.

  The smallest result is $8$ and the largest one $27$.

  Solution

  Here you see the expressions in the right order. Each time we use the quotient of powers properties for like terms.

  1. $\frac{2^5}{2^2}=2^{5-2}=2^3=8$
  2. $\frac{3^6}{3^4}=3^{6-4}=3^2=9$
  3. $\frac{4^7}{4^5}=4^{7-5}=4^2=16$
  4. $\frac{5^5}{5^3}=5^{5-3}=5^2=25$
  5. $\frac{3^5}{3^2}=3^{5-2}=3^3=27$

• Find the equivalent expressions.

  Hints

  Instead of dividing by a fraction you can multiply with the reciprocal of the fraction.

  Rearrange the fraction thus that you can use the quotient of powers property.

  Solution

  To use the quotient of powers property for like bases, we should first have to deal with what we already know about fractions. In the first problem that would be dividing by a fraction is the same as multiplying by its reciprocal.

  $\frac{21x^2y^3}{7xy^3}$

  • $\frac{21x^2y^3}{7xy^3}=\frac{21}{7}\cdot\frac{x^2}{x}\cdot\frac{y^3}{y^3}$
  • $\frac{21}7=3$
  • Now we use the quotient of powers property:
  • $\frac{x^2}{x}=x^{2-1}=x^1=x$
  • $\frac{y^3}{y^3}=y^{3-3}=y^{0}=1$

  Collecting all those results we get

  $\frac{21x^2y^3}{7xy^3}=3x$

  $\frac{x^2y^2}{5}\div\frac{x^2y}{25}=\frac{x^2y^2}{5}\cdot\frac{25}{x^2y}$

  • $\frac{25}5=5$
  • $\frac{x^2}{x^2}=x^{2-2}=x^0=1$
  • $\frac{y^2}{y}=y^{2-1}=y^{1}=y$

  Together we get

  $\frac{x^2y^2}{5}\div\frac{x^2y}{25}=5y$

  $\frac{10x^3y^2}{2xy^2}$

  • $\frac{10}2=5$
  • $\frac{x^3}{x}=x^{3-1}=x^2$
  • $\frac{y^2}{y^2}=y^{2-2}=y^0=1$

  Fine! So $\frac{10x^3y^2}{2xy^2}=5x^2$

  $\frac{1}{15xy}\div\frac{x^2}{75xy^2}$

  Once again we multiply with the reciprocal

  $\frac{1}{15xy}\div\frac{x^2}{75xy^2}=\frac{1}{15xy}\cdot\frac{75xy^2}{x^2}$

  and simplify first

  $\frac{y^2}{15x}\cdot\frac{75x^2}{y}=\frac{75x^2y^2}{15xy}$

  • $\frac{75}{15}=5$
  • $\frac{x^2}{x}=x^{2-1}=x^{1}=x$
  • $\frac{y^2}{y}=y^{2-1}=y^1=y$

  Thus we can conclude

  $\frac{y^2}{15x}\cdot\frac{75x^2}{y}=5xy$

• Complete the equations.

  Hints

  Keep $a^1=a$ in mind.

  Pay attention to the order of subtraction.

  Using this given example you can see that given factors cancel out.

  Solution

  You can determine the quotient of powers with like bases canceling as many factors as possible or using the quotient of powers property.

  The result is always the same.

  Canceling

  $\frac{a^2}a=\frac{a\cdot \not a}{\not a}=\frac{a}1=a$

  The quotient of powers property

  $\frac{a^2}a=a^{2-1}=a^1=a$ $~~~~$✓

• Rewrite the expressions in their simplest form.

  Hints

  Keep the following equations in mind:

  • $a^1=a$
  • $a^0=1$

  Use the quotient of powers property for like bases.

  Dividing by a fraction is the same as multiplying by its reciprocal.

  Solution

  $\mathbf{\frac{2^6 a^7}{b^5}\cdot\frac{b^{10}}{2^4a^4}}$

  We handle each base as its own in alphabetical order:

  • $\frac{2^6}{2^4}=2^{6-4}=2^2=4$
  • $\frac{a^7}{a^4}=a^{7-4}=a^3$
  • $\frac{b^{10}}{b^5}=b^{10-5}=b^5$

  Putting all those factors together we get the result $4a^3b^5$

  $\mathbf{\frac{33 x^4y^2}{x}\div\frac{11}{y^2}}$

  First, we multiply with the reciprocal of the second fraction

  $\frac{33 x^4y^2}{x}\div\frac{11}{y^2}=\frac{33 x^4y^2}{x}\cdot\frac{y^2}{11}$

  • $\frac{33}{11}=3$
  • $\frac{x^4}{x}=x^{4-1}=x^3$
  • $\frac{y^2}{y^2}=y^{2-2}=y^0=1$

  So we get $3x^3$ as result.

  $\mathbf{\frac{3^4}{b^2}\cdot\frac{a^6b^5}{3 a^4}}$

  Here we have to use the quotient of powers property three times:

  • $\frac{3^4}{3}=3^{4-1}=3^3=27$
  • $\frac{a^6}{a^4}=a^{6-4}=a^2$
  • $\frac{b^5}{b^2}=b^{5-2}=b^3$

  Together we get the following result $27a^2b^3$.

  Last let's do the following task

  $\mathbf{25x^2y^3\div\left(5^2xy\right)}$

  The fraction bar is just another to write down a division exercise:

  $25x^2y^3\div\left(5^2xy\right)=\frac{25x^2y^3}{5^2xy}$

  • $\frac{25}{5^2}=5^{2-2}=5^0=1$
  • $\frac{x^2}x=x^{2-1}=x$
  • $\frac{y^3}{y}=y^{3-1}=y^2$

  Finally, we get $xy^2$ as the result.