Lesson Explainer: Laws of Exponents over the Real Numbers | Nagwa Lesson Explainer: Laws of Exponents over the Real Numbers | Nagwa

Lesson Explainer: Laws of Exponents over the Real Numbers Mathematics • Second 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 to calculate a power raised to a power over the real numbers.

Recall that to evaluate a power, we multiply the base by the exponent the number of times indicated. Let’s recap the definition below.

Definition: Evaluating a Power

For a power with base 𝑎, 𝑎, and exponent 𝑛, 𝑛, then 𝑎=𝑎×𝑎×𝑎××𝑎.times

By using our understanding of how to evaluate power, we can derive the rules of how to multiply and divide powers as well as the laws for a power raised to a power.

Let’s say we have two powers with base 𝑎, 𝑎, where the first is raised to the power of 𝑚 and the second is raised to the power of 𝑛, 𝑚, 𝑛. If we multiply the two powers together, we get 𝑎×𝑎=(𝑎×𝑎×𝑎××𝑎)×(𝑎×𝑎×𝑎××𝑎).timestimes

If we count the number of times 𝑎 is multiplied by itself, this is 𝑚 times plus 𝑛 times, which is a total of 𝑚+𝑛 times. This then gives us 𝑎×𝑎=𝑎, which is the product rule for exponents.

Now, let’s say we have two powers with base 𝑎, 𝑎0, where the first is raised to the power of 𝑚 and the second is raised to the power of 𝑛, 𝑚, 𝑛. If we divide the first power by the second power, we get 𝑎÷𝑎=(𝑎×𝑎×𝑎××𝑎)÷(𝑎×𝑎×𝑎××𝑎)=𝑎×𝑎×𝑎×𝑎×𝑎××𝑎𝑎×𝑎×𝑎××𝑎.timestimes

If we cancel common factors in the numerator and the denominator, we end up canceling all the 𝑎’s in the numerator, meaning we have 𝑚 lots of 𝑎 minus 𝑛 lots of 𝑎, leaving us with 𝑚𝑛 lots of 𝑎. In other words, we have =𝑎×𝑎×𝑎×𝑎×𝑎×𝑎𝑎×𝑎×𝑎××𝑎=𝑎×𝑎××𝑎.timestimestimes

This then gives us, 𝑎÷𝑎=𝑎, when 𝑚𝑛.

If we consider the case when 𝑛 is larger than 𝑚, so 𝑚<𝑛, then we can use the same approach, but with a higher power in the denominator, giving us 𝑎÷𝑎=(𝑎×𝑎×𝑎×𝑎)÷(𝑎×𝑎×𝑎×𝑎×𝑎×𝑎)=𝑎×𝑎×𝑎××𝑎𝑎×𝑎×𝑎×𝑎×𝑎×𝑎.timestimes

Canceling common factors, we get =𝑎×𝑎×𝑎××𝑎𝑎×𝑎×𝑎×𝑎×𝑎×𝑎=1𝑎×𝑎××𝑎=1𝑎.timestimestimes

If we then use the law for negative indices, we get =𝑎.

Therefore, 𝑎÷𝑎=𝑎 for all integer values of 𝑚 and 𝑛, which is the quotient rule for exponents.

Next, let’s say we have a power with a base of 𝑎𝑏, 𝑎, 𝑏, which is raised to the power of 𝑛, 𝑛. This gives us (𝑎𝑏)=𝑎𝑏×𝑎𝑏×𝑎𝑏×𝑎𝑏.times

If we rewrite this so that all of the 𝑎’s and all of the 𝑏’s are together, we get (𝑎𝑏)=(𝑎×𝑎×𝑎×𝑎)×(𝑏×𝑏×𝑏×𝑏).timestimes

Since the number of times 𝑎 is multiplied by itself is 𝑛 times and similarly the number of times 𝑏 is multiplied by itself is 𝑛 times, then we get (𝑎𝑏)=𝑎𝑏, which is the power of a product rule for exponents.

Similarly, let’s say we have a power with a base of 𝑎𝑏, 𝑎, 𝑏0, which is raised to the power of 𝑛, 𝑛. This gives us 𝑎𝑏=𝑎𝑏×𝑎𝑏×𝑎𝑏×𝑎𝑏.times

If we rewrite this as one fraction, with the 𝑎’s in the numerator and the 𝑏’s in the denominator, we get 𝑎𝑏=𝑎×𝑎×𝑎×𝑎𝑏×𝑏×𝑏×𝑏.timestimes

Since the number of times 𝑎 is multiplied by itself is 𝑛 times and similarly the number of times 𝑏 is multiplied by itself is 𝑛 times, then we get 𝑎𝑏=𝑎𝑏, which is the power of a quotient rule for exponents.

Lastly, if we have now only one power with base 𝑎, 𝑎, which is raised to the power of 𝑚 and then raised to another power 𝑛, 𝑚, 𝑛, we get the following: (𝑎)=(𝑎)×(𝑎)×(𝑎)×(𝑎)=(𝑎×𝑎×𝑎×𝑎)×(𝑎×𝑎×𝑎×𝑎)×(𝑎×𝑎×𝑎×𝑎)×(𝑎×𝑎×𝑎×𝑎).𝑛𝑚timestimestimestimestimeslotsoftimes

If we count the number of times 𝑎 is multiplied by itself, this is 𝑚 times multiplied by 𝑛 times, which is a total of 𝑚𝑛 times. This then gives us (𝑎)=𝑎, which is called the power rule for exponents.

Let’s summarize the laws in the rules below.

Rule: Laws of Exponents

The following are the rules of exponents with real bases and integer exponents:

  • The product rule: 𝑎×𝑎=𝑎, where 𝑎 and 𝑚,𝑛.
  • The quotient rule: 𝑎÷𝑎=𝑎, where 𝑎0, 𝑚, and 𝑛.
  • The power of a product rule: (𝑎𝑏)=𝑎𝑏, where 𝑎, 𝑏 and 𝑛.
  • The power of a quotient rule: 𝑎𝑏=𝑎𝑏, where 𝑎, 𝑏0, and 𝑛.
  • The power rule: (𝑎)=𝑎, where 𝑎 and 𝑚, 𝑛.

As we are working with bases with real numbers, then often we need to simplify expressions with square roots. As such, it is helpful to recall the following definition.

Definition: Simplifying Expressions with Squares and Square Roots

For an expression with a base 𝑎, where 𝑎>0,

  • 𝑎=𝑎;
  • 𝑎=𝑎.

In our first example, we will discuss how to use the product rule for exponents to simplify an expression with a real base.

Example 1: Simplifying an Expression Involving Multiplication of Positive Integer Powers in the Real Numbers

Fill in the blank: The expression 3×3×3 simplifies to .

Answer

To find what 3×3×3 is equal to, we need to simplify the expression. As we have powers with the same base being multiplied together, then we can use the product rule for exponents, which states that 𝑎×𝑎=𝑎, where 𝑎 and 𝑚, 𝑛.

Starting with 3×3, we can write this as 3×3. Then, applying the product rule, we get 3×3=3=3.

Substituting 3 back into the original expression, we get 3×3×3=3×3.

Again, we can apply the product rule, which gives us 3×3=3=3.

Lastly, to simplify 3, we can use the property 𝑎=𝑎, where 𝑎>0. To do so, we can write 3 as a product of powers of 2 using the product law in reverse. This gives us 3=3×3×3=3×3×3=27.

Therefore, 3×3×3 simplifies to 27.

In the next example, we will use the quotient rule for exponents to simplify an expression with a real base.

Example 2: Simplifying an Expression Involving Division of Positive Integer Powers in the Real Numbers

Which of the following is equal to 7÷7?

  1. 7
  2. 7
  3. 7
  4. 7
  5. 7

Answer

To find what 7÷7 is equal to, we need to simplify the expression. As we have two powers with the same base, where one is divided by the other, then we can use the quotient rule for exponents, which states that 𝑎÷𝑎=𝑎, where 𝑎0, 𝑚, and 𝑛.

Applying this rule with 𝑎=7, 𝑚=8, and 𝑛=6, we get 7÷7=7=7.

Comparing with the options given, we can see that this is the same as option D, which is the correct answer.

In the next example, we will consider how to use rules for exponents to simplify expressions with square roots in them, where one base is the negative of another.

Example 3: Simplifying an Expression Involving Division of Positive Integer Powers in the Real Numbers

What is the value of 3÷3?

Answer

To find the value of 3÷3, we need to first simplify the expression. One approach is to use the laws of exponents.

We can see that the bases are nearly the same for the two parts of the expression, except that the first base is the negative of the second base. Instead, we can write the expression as 3÷3=1×3÷3.

Using the power of a product rule for exponents, which states that (𝑎𝑏)=𝑎𝑏, where 𝑎, 𝑏 and 𝑛, we can apply this to 1×3, giving us 1×3÷3=(1)3÷3.

Since 3 and 3 now have the same base, we can apply the quotient rule for exponents, which states that 𝑎÷𝑎=𝑎, where 𝑎0 and 𝑚, 𝑛, giving us (1)3÷3=(1)3=(1)3.

We can now evaluate this, by calculating (1) and using 𝑎=𝑎 and the product rule for exponents to find 3, as follows: (1)3=1×3=3×3=33.

Therefore, the value of 3÷3 is 33.

As with mathematical operations in general, we can apply multiple operations in one expression and use the laws of exponents to simplify these. We usually apply the laws according to the order of operations. We will explore this further in the next example.

Example 4: Evaluating a Numerical Expression Involving Real Numbers Using the Law of Exponents

Simplify 5×55.

Answer

To simplify 5×55, we need to use the laws of exponents. To do so, it is helpful to apply the laws to the numerator first, which is 5×5.

To simplify 5×5, we can use the product rule for exponents, which states that 𝑎×𝑎=𝑎, where 𝑎, 𝑚, 𝑛, giving us 5×5=5=5.

Substituting this back into the expression, we get 55. To simplify further, we can use the quotient rule for exponents, which states that 𝑎÷𝑎=𝑎, where 𝑎0, 𝑚, and 𝑛, giving us 55=5=5.

We can use the rule 𝑎=𝑎 and the product rule to simplify this further, giving us 5=5×5=5×5=25.

Therefore, 5×55 simplified gives us 25.

Next, we will consider how to simplify expressions where there are negative exponents. Let’s recall the laws of zero and negative exponents.

Laws: Rules for Zero and Negative Exponents

  • The law for zero exponents: 𝑎=1, where 𝑎0.
  • The law for negative exponents: 𝑎=1𝑎𝑎=1𝑎,or where 𝑎0 and 𝑛.

Using the law for negative exponents, we can derive two other properties.

First, if we take 𝑎=1𝑎, where 𝑎0 and 𝑛, and multiply both sides by 𝑎, we get 𝑎=1𝑎𝑎×𝑎=1.

Second, if we apply the law for negative exponents to a power with base 𝑎𝑏, where 𝑎, 𝑏0 and exponent 𝑛, we get 𝑎𝑏=1.

By the power of a quotient rule, which states that 𝑎𝑏=𝑎𝑏, where 𝑎, 𝑏0, and 𝑛, we then get 𝑎𝑏=1.

If we multiply the numerator and denominator by 𝑏, we then get 𝑎𝑏=1×𝑏𝑏=𝑏×𝑏=𝑏𝑎.

If we then apply the power of a quotient rule again, we get 𝑎𝑏=𝑏𝑎.

Similarly, we can show that 𝑎𝑏=𝑏𝑎.

Let’s summarize these two rules.

Laws: Rules for Zero and Negative Exponents

  • The law of multiplicative inverses: 𝑎×𝑎=1, where 𝑎0 and 𝑛.
  • The power of a quotient rule for negative exponents: 𝑎𝑏=𝑏𝑎𝑎𝑏=𝑏𝑎,or where 𝑎, 𝑏0 and 𝑛.

In the next example, we will use the law for negative exponents and the laws of indices to evaluate an expression.

Example 5: Evaluating a Numerical Expression Involving Real Numbers Using the Law of Exponents

For 𝑎=154 and 𝑏=52, evaluate 𝑎𝑏.

Answer

To evaluate 𝑎𝑏, we must start by substituting 𝑎=154 and 𝑏=52. This gives us 𝑎𝑏=15452.

Next, we can use the power of a quotient rule for negative exponents to simplify 52, which states that 𝑎𝑏=𝑏𝑎𝑎𝑏=𝑏𝑎,or where 𝑎, 𝑏0 and 𝑛, giving us 52=25.

Substituting this back to the original expression, we have 𝑎𝑏=154×25.

Using the power of a quotient rule, which states that 𝑎𝑏=𝑎𝑏, where 𝑎, 𝑏0, and 𝑛, we get 154=154 and 25=25.

This then gives us 𝑎𝑏=154×25.

If we expand the powers, we get 𝑎𝑏=154×154×25×25×25.

Canceling common factors then gives us 𝑎𝑏=154×154×25×25×25𝑎𝑏=3×32×5=910.

Therefore, for 𝑎=154 and 𝑏=52, the value of 𝑎𝑏 is 910.

So far, we have considered how to use the laws of exponents to simplify expressions with real bases. Next, we will consider how to solve simple exponential equations using the laws of exponents. Recall that if two powers are equal, then if either the bases or the exponents are also equal, the following rules apply.

Rule: Equating Powers with the Same Base or the Same Exponent

  • If 𝑎=𝑎, where 𝑎[1,0,1], then 𝑚=𝑛.
  • If 𝑎=𝑏, then 𝑎=𝑏 when 𝑛[1,3,5,] and |𝑎|=|𝑏| when 𝑛[2,4,6,].

In the next example, we will discuss how to use the laws of exponents to solve simple exponential equations.

Example 6: Using the Laws of Exponents to Solve a Simple Exponential Equation

Find the value of 𝑥 in the equation 16×2=2.

Answer

To find the value of 𝑥 in the equation 16×2=2, we can use the laws of exponents to simplify the different parts of the equation and then use the rule that states that if 𝑎=𝑎, where 𝑎[1,0,1], then 𝑚=𝑛, to find the value of 𝑥.

In order to apply the rules of equal powers, we need to have two powers with either the same base or the same exponent. As here the unknown is in the exponents, we need to write each side of the equation as a power with the same base. Meaning, we need to write 16 as 2, giving us 2×2, which we can simplify using the product rule, which states that 𝑎×𝑎=𝑎, where 𝑎 and 𝑚, 𝑛.

This then gives us on the left-hand side 2×2=2=2.

Looking at the right-hand side, 2, we can simplify this using the power rule, which states that (𝑎)=𝑎, where 𝑎 and 𝑚, 𝑛.

This then gives us on the right-hand side 2=2=2.()×

Equating both sides, we get 2=2.

Since the bases are both equal, then the exponents must also be equal, giving us 13𝑥=2𝑥2.

Solving for 𝑥, we then get 13𝑥=2𝑥215𝑥=2𝑥3𝑥=15𝑥=5.

Therefore, the value of 𝑥 is 5 in the equation 16×2=2.

In this explainer, we have learned how to use the laws of exponents to simplify expressions with real numbers raised to integer powers. We have also learned how to solve simple exponential equations using the laws of powers. Let’s recap the key points.

Key Points

  • We can use the laws of exponents to simplify powers with real bases and integer powers. These rules state the following:
    • The product rule: 𝑎×𝑎=𝑎, where 𝑎 and 𝑚, 𝑛.
    • The quotient rule: 𝑎÷𝑎=𝑎, where 𝑎0, 𝑚, and 𝑛.
    • The power of a product rule: (𝑎𝑏)=𝑎𝑏, where 𝑎, 𝑏 and 𝑛.
    • The power of a quotient rule: 𝑎𝑏=𝑎𝑏, where 𝑎, 𝑏0, and 𝑛.
    • The power rule: (𝑎)=𝑎, where 𝑎 and 𝑚, 𝑛.
    • The law of multiplicative inverses: 𝑎×𝑎=1, where 𝑎0 and 𝑛.
    • The power of a quotient rule for negative exponents: 𝑎𝑏=𝑏𝑎𝑎𝑏=𝑏𝑎,or where 𝑎, 𝑏0 and 𝑛.
  • To solve simple exponential equations with real bases, we can use the following rules when the bases or exponents are equal:
    • If 𝑎=𝑎, where 𝑎[1,0,1], then 𝑚=𝑛.
    • If 𝑎=𝑏, then 𝑎=𝑏 when 𝑛[1,3,5,] and |𝑎|=|𝑏| when 𝑛[2,4,6,].

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