Learning Outcomes
- Simplify expressions with negative exponents
The Quotient Property of Exponents, introduced in Divide Monomials, had two forms depending on whether the exponent in the numerator or denominator was larger.
Question: Find The Lengths Of The Sides Of The Triangle PQR.(a) P(5, 0,?1), Q(9, 2, 3), R(3, 4, 3) PQ = QR = RP = Is It A Right Triangle?YesNo Is It An. Start studying given:Q is the midpoint of PR, R is the midpoint of OS Prove:PQ= 1/2 QS. Learn vocabulary, terms, and more with flashcards, games, and other study tools.
Quotient Property of Exponents
In this type of right triangle, the sides corresponding to the angles 30°-60°-90° follow a ratio of 1:√ 3:2. Thus, in this type of triangle, if the length of one side and the side's corresponding angle is known, the length of the other sides can be determined using the above ratio. In the fig, PQR is a triangle, right angled at Q. If XY QR, PQ = 6cm, PY = 4cm & PX: XQ = 1: 2 Calculate the lengths of PR and QR. Weekly Subscription $1.99 USD per week until cancelled Monthly Subscription $4.99 USD per month until cancelled Annual Subscription $29.99 USD per year until cancelled.
If [latex]a[/latex] is a real number, [latex]ane 0[/latex], and [latex]m,n[/latex] are whole numbers, then
[latex]frac{{a}^{m}}{{a}^{n}}={a}^{m-n},m>ntext{and}frac{{a}^{m}}{{a}^{n}}=frac{1}{{a}^{n-m}},n>m[/latex]
What if we just subtract exponents, regardless of which is larger? Let’s consider [latex]frac{{x}^{2}}{{x}^{5}}[/latex].
We subtract the exponent in the denominator from the exponent in the numerator.
We subtract the exponent in the denominator from the exponent in the numerator.
[latex]frac{{x}^{2}}{{x}^{5}}[/latex]
[latex]{x}^{2 - 5}[/latex]
[latex]{x}^{-3}[/latex]
We can also simplify [latex]frac{{x}^{2}}{{x}^{5}}[/latex] by dividing out common factors: [latex]frac{{x}^{2}}{{x}^{5}}[/latex].
[latex]{x}^{2 - 5}[/latex]
[latex]{x}^{-3}[/latex]
We can also simplify [latex]frac{{x}^{2}}{{x}^{5}}[/latex] by dividing out common factors: [latex]frac{{x}^{2}}{{x}^{5}}[/latex].
This implies that [latex]{x}^{-3}=frac{1}{{x}^{3}}[/latex] and it leads us to the definition of a negative exponent.
Negative Exponent
If [latex]n[/latex] is a positive integer and [latex]ane 0[/latex], then [latex]{a}^{-n}=frac{1}{{a}^{n}}[/latex].
The negative exponent tells us to re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents.
example
Simplify:
1. [latex]{4}^{-2}[/latex]
2. [latex]{10}^{-3}[/latex]
2. [latex]{10}^{-3}[/latex]
Solution
1. | |
[latex]{4}^{-2}[/latex] | |
Use the definition of a negative exponent, [latex]{a}^{-n}=frac{1}{{a}^{n}}[/latex]. | [latex]frac{1}{{4}^{2}}[/latex] |
Simplify. | [latex]frac{1}{16}[/latex] |
![Right qr 1 2s Right qr 1 2s](https://www.qrcode.com/en/img/model12/model2Image.png)
2. | |
[latex]{10}^{-3}[/latex] | |
Use the definition of a negative exponent, [latex]{a}^{-n}=frac{1}{{a}^{n}}[/latex]. | [latex]frac{1}{{10}^{3}}[/latex] |
Simplify. | [latex]frac{1}{1000}[/latex] |
When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.
example
Simplify:
1. [latex]{left(-3right)}^{-2}[/latex]
2 [latex]{-3}^{-2}[/latex]
2 [latex]{-3}^{-2}[/latex]
Right Qr 1 2 X 4
Show SolutionSolution
The negative in the exponent does not affect the sign of the base.
The negative in the exponent does not affect the sign of the base.
1. | |
The exponent applies to the base, [latex]-3[/latex] . | [latex]{left(-3right)}^{-2}[/latex] |
Take the reciprocal of the base and change the sign of the exponent. | [latex]frac{1}{{left(-3right)}^{2}}[/latex] |
Simplify. | [latex]frac{1}{9}[/latex] |
2. | |
The expression [latex]-{3}^{-2}[/latex] means “find the opposite of [latex]{3}^{-2}[/latex] “. The exponent applies only to the base, [latex]3[/latex]. | [latex]-{3}^{-2}[/latex] |
Rewrite as a product with [latex]−1[/latex]. | [latex]-1cdot {3}^{-2}[/latex] |
Take the reciprocal of the base and change the sign of the exponent. | [latex]-1cdot frac{1}{{3}^{2}}[/latex] |
Simplify. | [latex]-frac{1}{9}[/latex] |
We must be careful to follow the order of operations. In the next example, parts 1 and 2 look similar, but we get different results.
example
Simplify:
1. [latex]4cdot {2}^{-1}[/latex]
2. [latex]{left(4cdot 2right)}^{-1}[/latex] Locko 1 0 – password manager and file vault.
Show Solution2. [latex]{left(4cdot 2right)}^{-1}[/latex] Locko 1 0 – password manager and file vault.
Solution
Remember to always follow the order of operations.
Remember to always follow the order of operations.
1. | |
Do exponents before multiplication. | [latex]4cdot {2}^{-1}[/latex] |
Use [latex]{a}^{-n}=frac{1}{{a}^{n}}[/latex]. | [latex]4cdot frac{1}{{2}^{1}}[/latex] |
Simplify. | [latex]2[/latex] |
2. | [latex]{left(4cdot 2right)}^{-1}[/latex] |
Simplify inside the parentheses first. | [latex]{left(8right)}^{-1}[/latex] |
Use [latex]{a}^{-n}=frac{1}{{a}^{n}}[/latex]. | [latex]frac{1}{{8}^{1}}[/latex] |
Simplify. | [latex]frac{1}{8}[/latex] |
When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers.
example
Simplify: [latex]{x}^{-6}[/latex].
Show Solution[latex]{x}^{-6}[/latex] | |
Use the definition of a negative exponent, [latex]{a}^{-n}=frac{1}{{a}^{n}}[/latex]. | [latex]frac{1}{{x}^{6}}[/latex] |
When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the order of operations, expressions in parentheses are simplified before exponents are applied. We’ll see how this works in the next example.
example
Simplify:
1. [latex]5{y}^{-1}[/latex]
2. [latex]{left(5yright)}^{-1}[/latex]
3. [latex]{left(-5yright)}^{-1}[/latex]
Show Solution2. [latex]{left(5yright)}^{-1}[/latex]
3. [latex]{left(-5yright)}^{-1}[/latex]
Solution
1. | |
Notice the exponent applies to just the base [latex]y[/latex] . | [latex]5{y}^{-1}[/latex] |
Take the reciprocal of [latex]y[/latex] and change the sign of the exponent. | [latex]5cdot frac{1}{{y}^{1}}[/latex] |
Simplify. | [latex]frac{5}{y}[/latex] |
Right Qr 1 2 0
2. | |
Here the parentheses make the exponent apply to the base [latex]5y[/latex] . | [latex]{left(5yright)}^{-1}[/latex] |
Take the reciprocal of [latex]5y[/latex] and change the sign of the exponent. | [latex]frac{1}{{left(5yright)}^{1}}[/latex] |
Simplify. | [latex]frac{1}{5y}[/latex] |
3. | |
[latex]{left(-5yright)}^{-1}[/latex] | |
The base is [latex]-5y[/latex] . Take the reciprocal of [latex]-5y[/latex] and change the sign of the exponent. | [latex]frac{1}{{left(-5yright)}^{1}}[/latex] |
Simplify. | [latex]frac{1}{-5y}[/latex] |
Use [latex]frac{a}{-b}=-frac{a}{b}[/latex]. | [latex]-frac{1}{5y}[/latex] |
Right Qr 1 2s
VIDEO REQUEST
Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form, [latex]frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex], where [latex]ane 0[/latex] and m and n are integers.
Right Qr 1 2 3
When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, we will rewrite it by using the definition of negative exponents, [latex]{a}^{-n}=frac{1}{{a}^{n}}[/latex].