Dividing with Exponents

[One speaker]

[Video is a presentation that starts with the words “Rules of Exponents: Dividing with Exponents or Quotient Rule.”]

Narrator: Hi, welcome to the video on the rules of exponents when we’re dividing with exponents, or in other words the quotient rule. A quotient is what you get when you divide one number by another number [shows the definition].

Here’s an example: x to the fourth divided by x cubed [writes the equation: x⁴/x³]. If we write this out, this is the same as x multiplied together four times in the numerator, divided by x multiplied together three times in the denominator [writes the equation: xxxx / xxx]. Now, remember, anything divided by itself, such as like a divided by a equals 1 [writes the equation: a/a=1]. So, up here [points to the equation] we have an x divided by itself, another x, and we see this two other times [circles in the pairs of x’s]. So in essence this can be rewritten as one times one times one times x divided by one [writes: 1∙1∙1∙x/1] because this is all being multiplied by one. Because anything multiplied by one is still itself. But x divided by 1 [points to x/1] is also still itself.  One times one times one times x is equal to x  [writes: =x].

This demonstrates the rule for when you have terms with the same base, in this case both were x, and you’re finding the quotient, or in other words, you’re dividing one from another. What happens is you get the same as x to the fourth minus three [writes the equation: x⁴-³].  In other words, x to the exponent of the term in the numerator, minus the exponent of the term in the denominator, which equals x to the one power [writes: x¹], four minus three is one, which is just the same as x [writes: =x]. X to the one power is just x.

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Our rules state that if you have terms, with the same base such as in this case, we’re going to use the base of a [writes a/a], raised to different powers, let’s say to the x and raised to the y [adds exponents leaving the equation like: aˣ/aʸ]  You’re finding the quotient, or you’re dividing these, it’s equal to a to the x minus y [writes: aˣ-ʸ]. Notice, when we were multiplying together, if we had a to x, times a to the y it equaled a to the x plus y [writes the equation: aˣ aʸ = aˣ+ʸ]. But when we’re dividing it’s equal to a to the x minus y [points to the previous answer]. Let’s look at another example.

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M to the sixth power divided by m to the two power is equal to six m’s on top divided by two m’s in the denominator [writes the equation: m⁶/m² = m m m m m m/m m]. Since everything is being multiplied and divided, we can cancel out two of our m’s [crosses two m’s in the numerator and two in the denominator], because m divided by m is one, and m divided by m is one, leaving us with just four m’s in the numerator. Which equals m to the fourth [writes: =m⁴]. Note, this is the same as m to the sixth minus two which equals m to the fourth [writes the equation: m⁶-²=m⁴]. So we got the same answer.

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Now, let’s do one more example where we have two different variables, or in other words terms with different bases. We have both x and m as bases [writes the equation: x²m³x³/m²x]. We can combine the terms with the same base. So this is equal to x to the two plus three, in the numerator, times m to the three power, then divided by m to the two power timex x [writes: =x²+³m³/m²x¹], and this x [pointing to the denominator] is just to the one power. If it’s the x all to itself it just means to the one power. So now we can simplify this a little bit, x to the two plus three is x to the five times m cubed, divided by m squared times x [writes: =x⁵m³/m²x].

Now we can use our rule for division where we subtract exponents. So we can rewrite this as x to the five minus one [writes: x⁵-¹], since this is a one exponent on this x [x in the denominator] . So, x to the five minus one times m to three minus two [adds: m³-²], which equals x to the fourth, m to 1 [writes: =x⁴m¹], or in other words, just m. [points to the entire equation: x⁵-¹ m³-² = x⁴m¹]

So, x to the fourth times m is our final answer [circles the answer].      

[End of video.]