Lesson Explainer: Laws of Exponents | Nagwa Lesson Explainer: Laws of Exponents | Nagwa

Lesson Explainer: Laws of Exponents Mathematics • First Year of Preparatory School

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In this explainer, we will learn how to apply the laws of exponents to multiply and divide powers and work out a power raised to a power.

We begin by recalling that repeated multiplication is represented by exponentiation. In general, we say that raising a number to a positive integer exponent 𝑛 is the same as multiplying it by itself such that it appears 𝑛 times in the product. If we have a base of 𝑎𝑏, then we have the following: 𝑎𝑏=𝑎𝑏×𝑎𝑏××𝑎𝑏=𝑎×𝑎××𝑎𝑏×𝑏××𝑏=𝑎𝑏.times

This definition and result allow us to prove some useful results in evaluating exponential expressions. We call these the laws of exponents.

First, we can consider multiplying two exponential expressions with the same base. For example, 2×2. We have 2×2=(2×2×2)×(2×2×2×2)=2×2×2×2×2×2×2=2.()timestimestimes

This suggests that when we multiply exponential expressions with the same base, we add the exponents. We can show that this result holds in general by following the same process.

Let’s say that 𝑎𝑏 and 𝑛 and 𝑚 are positive integers; then, 𝑎𝑏×𝑎𝑏=𝑎𝑏×𝑎𝑏××𝑎𝑏×𝑎𝑏×𝑎𝑏××𝑎𝑏.timestimes

Remember we can evaluate a product of rational numbers in any order, so we can rewrite this as a product of 𝑎𝑏 with itself where it appears 𝑛+𝑚 times. Hence, 𝑎𝑏×𝑎𝑏=𝑎𝑏×𝑎𝑏××𝑎𝑏×𝑎𝑏×𝑎𝑏××𝑎𝑏=𝑎𝑏×𝑎𝑏××𝑎𝑏=𝑎𝑏.timestimestimes

We can follow a similar process if we are dividing two exponential expressions with the same base. However, we also need the exponent of the dividend to be greater than the exponent of the divisor.

For example, if we want to calculate 23÷23, we have 23÷23=23÷23.

Instead of dividing, we multiply by the reciprocal to get 23÷23=23×32.

We then write out the exponents in full and cancel out the shared factors:

Since we cancel out four of the factors of 2 and 3 from the powers of 5, we can think of this as 23. This then suggests that when we divide exponential expressions with the same base, we subtract the exponents. We can show that this result holds for any rational number by following the same process.

If 𝑎𝑏 and 𝑛 and 𝑚 are positive integers with 𝑛>𝑚, we have 𝑎𝑏÷𝑎𝑏=𝑎𝑏÷𝑎𝑏=𝑎𝑏×𝑏𝑎.

We can rearrange by writing the products out in full to get

Since 𝑛>𝑚, we can cancel out 𝑚 of the shared factors of 𝑎 in the first fraction and 𝑚 of the shared factors of 𝑏 in the second fraction. This will leave 𝑛𝑚 factors of 𝑎 in the numerator and 𝑛𝑚 factors of 𝑏 in the denominator. Hence, 𝑎𝑏÷𝑎𝑏=(𝑎×𝑎××𝑎)1×1(𝑏×𝑏××𝑏)=𝑎×1𝑏=𝑎𝑏.factorsfactors

Finally, we can rewrite this as an exponent of 𝑎𝑏. We have 𝑎𝑏÷𝑎𝑏=𝑎𝑏.

There is one last rule we can deduce by using this definition. We can consider what would happen if we raised a base to multiple exponents.

For example, let’s evaluate 4. We start with the expression inside the parentheses, so 4=(4×4).

We then need to cube this expression. So, we get (4×4)=(4×4)×(4×4)×(4×4).

We see that this is 4 multiplied by itself, where it appears 6 times in the product. However, we can see where this value of 6 comes from. Since we cube 4×4, we multiply the number of factors by 3, which gives us 2×3=6 factors of 4. Hence, we have 4=(4×4)×(4×4)×(4×4)=4.(×)

We can show that this result holds for any rational number by following the same process.

If 𝑎𝑏 and 𝑛 and 𝑚 are positive integers, then

We can summarize all the results we have just proven as follows.

Rules: Laws of Exponents for Rational Bases

If 𝑎𝑏 and 𝑛 and 𝑚 are positive integers, then

  1. 𝑎𝑏=𝑎𝑏,
  2. 𝑎𝑏×𝑎𝑏=𝑎𝑏,
  3. 𝑎𝑏=𝑎𝑏×,
  4. and if 𝑛>𝑚, then
    𝑎𝑏÷𝑎𝑏=𝑎𝑏.

Let’s now see some examples of using the laws of exponents to evaluate and simplify exponential expressions.

Example 1: Multiplying Negative Mixed Number Expressions with Positive Integer Exponents

Which of the following is equal to 112×112?

  1. 71932
  2. 71932
  3. 32243
  4. 31251024
  5. 243

Answer

We could evaluate this expression by evaluating each factor separately; however, we can also simplify the expression first by using the laws of exponents. We recall that if we multiply two powers with the same base, then we can just add the exponents, 𝑏×𝑏=𝑏, for any rational number 𝑏 and positive integers 𝑛 and 𝑚.

So, 112×112=112=112.

We can evaluate this expression by first rewriting 112 as a fraction. We note that 112=32. Thus, 112=32.

Finally, we recall that if 𝑎𝑏 is a rational number and 𝑛 is a positive integer, then 𝑎𝑏=𝑎𝑏.

Hence, 32=(3)2=24332.

We could leave our answer like this; however, it is not one of the options. Instead, we rewrite this as a mixed number. We have 24332=32×7+1932=71932, which is option B.

In our next example, we will use a different one of the laws of exponents to evaluate an expression.

Example 2: Dividing Negative Mixed Number Expressions by Positive Integer Exponents

Which of the following has the same value as 112÷112?

  1. 23
  2. 23
  3. 32
  4. 32
  5. 24332

Answer

We could evaluate the dividend and divisor separately and then evaluate the division. However, it is easier to simplify the expression first. We recall that the laws of exponents tell us that when dividing powers with the same base, we subtract the exponents, 𝑏÷𝑏=𝑏, for any rational number 𝑏 and positive integers 𝑛 and 𝑚 with 𝑛>𝑚.

Thus, 112÷112=112=112.

We then recall that raising a number to an exponent of 1 leaves it unchanged, so 112=112.

Finally, we convert our answer into a fraction. We have 112=1×2+12=32, which is option D.

In our next example, we will determine which of a set of given options is equivalent to a given exponential expression.

Example 3: Evaluating Positive Rational Expressions with Positive Integer Exponents Raised to Positive Integer Exponents

Which of the following expressions has the same value as 13?

  1. 13
  2. 13
  3. 13
  4. 23
  5. 13×13

Answer

We could evaluate this expression and all of the options by just evaluating the exponents. However, it is easier to simplify the given expression by using the laws of exponents.

We recall that if we raise a rational base to a positive integer exponent and then raise this to another positive integer exponent, then we can simplify by multiplying the exponents, (𝑏)=𝑏,× for any rational number 𝑏 and positive integers 𝑛 and 𝑚.

Therefore, 13=13=13.×

We see that this is option A.

In our next example, we will use the laws of exponents to simplify the evaluation of a given exponential expression.

Example 4: Evaluating a Numerical Expression Using the Laws of Exponents

Calculate 3×1×, giving your answer in its simplest form.

Answer

We could evaluate this expression by evaluating each power in turn and then multiplying the fractions. However, this is a difficult process and it would be easy to make mistakes. Instead, we can simplify by using the laws of exponents.

First, we rewrite each mixed number as a fraction. We have 315=15515=165,112=2212=32.

This allows us to rewrite the expression as 3×1×=××.

We can now see that the expression contains the quotients of the same bases raised to positive integer powers. We can rewrite the expression to highlight taking the quotients of the same bases: ××=×.

We now recall that for any rational number 𝑎𝑏 and positive integer exponents 𝑛 and 𝑚 such that 𝑛>𝑚, we have 𝑎𝑏÷𝑎𝑏=𝑎𝑏.

Thus, ×=165×32=165×32.

We know that raising a number to first power leaves it unchanged. We can also distribute the exponent of 2 over the fraction by recalling that 𝑎𝑏=𝑎𝑏 for any rational number 𝑎𝑏 and positive integer 𝑛. This allows us to rewrite the expression as follows: 165×32=165×(3)2.

We can now note that (3)=9 and 2=4, allowing us to rewrite this as 165×(3)2=165×94.

We can now evaluate the product 165×94=16×95×4=365.

In our next example, we will evaluate an algebraic expression using the laws of exponents.

Example 5: Evaluating an Algebraic Expression Using the Laws of Exponents

If 𝑥=15, 𝑦=12, and 𝑧=25, find the value of 𝑥÷(𝑦𝑧).

Answer

We first substitute the given values into the expression. This gives us 𝑥÷(𝑦𝑧)=15÷1225.

We can simplify this expression by evaluating the product inside the square as follows: 1225=1×22×5=15.

Thus, 15÷1225=15÷15.

We can now note that we are dividing exponential expressions with the same rational base. We can recall that the laws of exponents tell us that when dividing exponential expressions with the same rational base, we subtract the exponents, 𝑏÷𝑏=𝑏, for any rational number 𝑏 and positive integers 𝑛 and 𝑚 with 𝑛>𝑚.

Hence, 15÷15=15=15.

Finally, we evaluate the exponent: 15=15=125.

In our final example, we will simplify an algebraic expression using the laws of exponents.

Example 6: Simplifying an Algebraic Expression Using the Laws of Exponents

Given that x is a rational number, which of the following expressions is equivalent to 𝑥×𝑥+𝑥?

  1. 𝑥+𝑥
  2. 𝑥+𝑥
  3. 𝑥+𝑥
  4. 𝑥+𝑥
  5. 𝑥𝑥+𝑥

Answer

Since 𝑥 is a rational number, we can apply the laws of exponents to simplify this algebraic expression. We note that the expression contains the product of exponential expressions with the same rational base and the repeated exponentiation of a rational base. So, we recall the two laws of exponents 𝑏×𝑏=𝑏,(𝑏)=𝑏,× for any rational number 𝑏 and positive integers 𝑛 and 𝑚.

Thus, 𝑥×𝑥=𝑥=𝑥 and 𝑥=𝑥=𝑥.×

We can then substitute these into the expression to get 𝑥×𝑥+𝑥=𝑥+𝑥, which we can see is option A.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • If 𝑎𝑏 and 𝑛 is a positive integer, then 𝑎𝑏=𝑎𝑏.
  • If 𝑎𝑏 and 𝑛 and 𝑚 are positive integers, then 𝑎𝑏×𝑎𝑏=𝑎𝑏.
  • If 𝑎𝑏 and 𝑛 and 𝑚 are positive integers with 𝑛>𝑚, then 𝑎𝑏÷𝑎𝑏=𝑎𝑏.
  • If 𝑎𝑏 and 𝑛 and 𝑚 are positive integers, then 𝑎𝑏=𝑎𝑏.×
  • We can use the laws of exponents to simplify exponential expressions.

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