Applying Combined Exponents Rules - Examples, Exercises and Solutions

Question Types:
Applying Combined Exponents Rules: Applying the formulaApplying Combined Exponents Rules: Calculating powers with negative exponentsApplying Combined Exponents Rules: Complete the equationApplying Combined Exponents Rules: converting Negative Exponents to Positive ExponentsApplying Combined Exponents Rules: Factoring Out the Greatest Common Factor (GCF)Applying Combined Exponents Rules: FactorizationApplying Combined Exponents Rules: Identify the greater valueApplying Combined Exponents Rules: More than one unknownApplying Combined Exponents Rules: Multiplying Exponents with the same baseApplying Combined Exponents Rules: Number of termsApplying Combined Exponents Rules: A power lawApplying Combined Exponents Rules: Presenting powers in the denominator as powers with negative exponentsApplying Combined Exponents Rules: Presenting powers with negative exponents as fractionsApplying Combined Exponents Rules: Presenting powers with negative exponents as fractionsApplying Combined Exponents Rules: MonomialApplying Combined Exponents Rules: Single VariableApplying Combined Exponents Rules: TrinomialApplying Combined Exponents Rules: BinomialApplying Combined Exponents Rules: Two VariablesApplying Combined Exponents Rules: Using laws of exponents with parametersApplying Combined Exponents Rules: Using multiple rulesApplying Combined Exponents Rules: Using the laws of exponentsApplying Combined Exponents Rules: Using variablesApplying Combined Exponents Rules: Variable in the base of the powerApplying Combined Exponents Rules: Variable in the exponent of the powerApplying Combined Exponents Rules: Variables in the exponent of the powerApplying Combined Exponents Rules: Worded problemsApplying Combined Exponents Rules: The power of zero

Taking advantage of all the properties of powers or laws of exponents

From time to time, we will come across exercises in which we must use all the properties of powers together.
As soon as you have the exercise, try to first get rid of the parentheses according to the properties of powers and then, apply these properties to the corresponding terms, one after the other.

All the properties of powers or laws of exponents are:
am×an=a(m+n)a^m\times a^n=a^{(m+n)}
aman=a(mn)\frac {a^m}{a^n} =a^{(m-n)}
(a×b)n=an×bn(a\times b)^n=a^n\times b^n
(ab)n=anbn(\frac {a}{b})^n=\frac {a^n}{b^n}
(an)m=a(nm)(a^n )^m=a^{(n*m)}
a0=1a^0=1
When a0a≠0
an=1ana^{-n}=\frac {1}{a^n}

Suggested Topics to Practice in Advance

  1. Multiplying Exponents with the Same Base
  2. Division of Exponents with the Same Base
  3. Exponent of a Multiplication
  4. Power of a Quotient
  5. Power of a Power

Practice Applying Combined Exponents Rules

Examples with solutions for Applying Combined Exponents Rules

Exercise #1

1120=? 112^0=\text{?}

Video Solution

Step-by-Step Solution

We use the zero exponent rule.

X0=1 X^0=1 We obtain

1120=1 112^0=1 Therefore, the correct answer is option C.

Answer

1

Exercise #2

Solve the following problem:

13= 1^3=

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify the given information

  • Step 2: Apply the appropriate exponent rule

  • Step 3: Perform the calculation

Now, let's work through each step:
Step 1: The problem gives us the expression 13 1^3 . This means we have a base of 1 and an exponent of 3.
Step 2: We'll use the exponentiation rule, which states that an=a×a××a a^n = a \times a \times \ldots \times a (n times).
Step 3: Since our base is 1, raising 1 to any power will still result in 1. Therefore, we can express this as 1×1×1=1 1 \times 1 \times 1 = 1 .

Therefore, the solution to 13 1^3 is 1 1 .

Answer

1 1

Exercise #3

Solve the following problem:

70= 7^0=

Video Solution

Step-by-Step Solution

To solve the problem of finding 70 7^0 , we will follow these steps:

  • Step 1: Identify the general rule for exponents with zero.

  • Step 2: Apply the rule to the given problem.

  • Step 3: Consider the provided answer choices and select the correct one.

Now, let's work through each step:

Step 1: A fundamental rule in exponents is that any non-zero number raised to the power of zero is equal to one. This can be expressed as: a0=1 a^0 = 1 where a a is not zero.

Step 2: Apply this rule to the problem: Since we have 70 7^0 , and 7 7 is certainly a non-zero number, the expression evaluates to 1. Therefore, 70=1 7^0 = 1 .

Therefore, the solution to the problem is 70=1 7^0 = 1 , which corresponds to choice 2.

Answer

1 1

Exercise #4

2423= \frac{2^4}{2^3}=

Video Solution

Step-by-Step Solution

Let's keep in mind that the numerator and denominator of the fraction have terms with the same base, therefore we use the property of powers to divide between terms with the same base:

bmbn=bmn \frac{b^m}{b^n}=b^{m-n} We apply it in the problem:

2423=243=21 \frac{2^4}{2^3}=2^{4-3}=2^1 Remember that any number raised to the 1st power is equal to the number itself, meaning that:

b1=b b^1=b Therefore, in the problem we obtain:

21=2 2^1=2 Therefore, the correct answer is option a.

Answer

2 2

Exercise #5

3532= \frac{3^5}{3^2}=

Video Solution

Step-by-Step Solution

Using the quotient rule for exponents: aman=amn \frac{a^m}{a^n} = a^{m-n} .

Here, we have 3532=352 \frac{3^5}{3^2} = 3^{5-2}

Simplifying, we get 33 3^3

Answer

33 3^3

Exercise #6

Reduce the following equation:

(32)4×(53)5= \left(3^2\right)^4\times\left(5^3\right)^5=

Video Solution

Step-by-Step Solution

To solve this problem, we'll employ the power of a power rule in exponents, which states that (am)n=am×n(a^m)^n = a^{m \times n}.

Let's apply this rule to each part of the expression:

  • Step 1: Simplify (32)4(3^2)^4
    According to the power of a power rule, this becomes 32×4=383^{2 \times 4} = 3^8.

  • Step 2: Simplify (53)5(5^3)^5
    Similarly, apply the rule here to get 53×5=5155^{3 \times 5} = 5^{15}.

After simplifying both parts, we multiply the results:

38×5153^8 \times 5^{15}

Thus, the reduced expression is 38×515\boxed{3^8 \times 5^{15}}.

Answer

38×515 3^8\times5^{15}

Exercise #7

Solve the following problem:

(34)×(32)= \left(3^4\right)\times\left(3^2\right)=

Video Solution

Step-by-Step Solution

In order to solve this problem, we'll follow these steps:

  • Step 1: Identify the base and exponents

  • Step 2: Use the formula for multiplying powers with the same base

  • Step 3: Simplify the expression by applying the relevant exponent rule

Now, let's work through each step:

Step 1: The given expression is (34)×(32) (3^4) \times (3^2) . Here, the base is 3, and the exponents are 4 and 2.

Step 2: Apply the exponent rule, which states that when multiplying powers with the same base, we add the exponents:
am×an=am+n a^m \times a^n = a^{m+n}

Step 3: Using the rule identified in Step 2, we add the exponents 4 and 2:
34×32=34+2=36 3^4 \times 3^2 = 3^{4+2} = 3^6

Therefore, the simplified form of the expression is 36 3^6 .

Answer

36 3^6

Exercise #8

Solve the following problem:

(3)0= \left(-3\right)^0=

Video Solution

Step-by-Step Solution

To solve this problem, let's follow these steps:

  • Understand the zero exponent rule.

  • Apply this rule to the given expression.

  • Identify the correct answer from the given options.

According to the rule of exponents, any non-zero number raised to the power of zero is equal to 11. This is one of the fundamental properties of exponents.
Now, apply this rule:

Step 1: We are given the expression (3)0(-3)^0.
Step 2: Here, 3-3 is our base. We apply the zero exponent rule, which tells us that (3)0=1(-3)^0 = 1.

Therefore, the value of (3)0(-3)^0 is 11.

Answer

1 1

Exercise #9

(35)4= (3^5)^4=

Video Solution

Step-by-Step Solution

To solve the exercise we use the power property:(an)m=anm (a^n)^m=a^{n\cdot m}

We use the property with our exercise and solve:

(35)4=35×4=320 (3^5)^4=3^{5\times4}=3^{20}

Answer

320 3^{20}

Exercise #10

Simplify the following equation:

42×35×43×32= 4^2\times3^5\times4^3\times3^2=

Video Solution

Step-by-Step Solution

To simplify the given expression 42×35×43×32 4^2 \times 3^5 \times 4^3 \times 3^2 , we will follow these steps:

  • Step 1: Identify and group similar bases.

  • Step 2: Apply the rule for multiplying like bases.

  • Step 3: Simplify the expression.

Now, let's go through each step thoroughly:

Step 1: Identify and group similar bases:
We see two distinct bases here: 4 and 3.

Step 2: Apply the rule for multiplying like bases:
For base 4: Combine 424^2 and 434^3, using the rule am×an=am+na^m \times a^n = a^{m+n}.

Add the exponents for base 4: 2+3=5 2 + 3 = 5 , thus, 42×43=45 4^2 \times 4^3 = 4^5 .

For base 3: Combine 353^5 and 323^2, still using the same exponent rule.

Add the exponents for base 3: 5+2=7 5 + 2 = 7 , resulting in 35×32=37 3^5 \times 3^2 = 3^7 .

Step 3: Simplify the expression:
The simplified expression is 45×37 4^5 \times 3^7 .

Therefore, the final simplified expression is 45×37 4^5 \times 3^7 .

Answer

45×37 4^5\times3^7

Exercise #11

Simplify the following equation:

47×53×42×54= 4^7\times5^3\times4^2\times5^4=

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Identify and group the terms with the same base.

  • Step 2: Apply the laws of exponents to simplify by adding the exponents of each base.

  • Step 3: Write the simplified form.

Let's work through each step:

Step 1: We are given that 47×53×42×54 4^7 \times 5^3 \times 4^2 \times 5^4 .

Step 2: First, group the terms with the same base:

47×42 4^7 \times 4^2 and 53×54 5^3 \times 5^4 .

Step 3: Use the law of exponents, which states am×an=am+n a^m \times a^n = a^{m+n} .

For the base 4: 47×42=47+2=49 4^7 \times 4^2 = 4^{7+2} = 4^9 .

For the base 5: 53×54=53+4=57 5^3 \times 5^4 = 5^{3+4} = 5^7 .

Therefore, the simplified form of the expression is 49×57 4^9 \times 5^7 .

Answer

49×57 4^9\times5^7

Exercise #12

5654= \frac{5^6}{5^4}=

Video Solution

Step-by-Step Solution

Using the quotient rule for exponents: aman=amn \frac{a^m}{a^n} = a^{m-n} .

Here, we have 5654=564 \frac{5^6}{5^4} = 5^{6-4} . Simplifying, we get 52 5^2 .

Answer

52 5^2

Exercise #13

Simplify the following equation:

53×24×52×23= 5^3\times2^4\times5^2\times2^3=

Video Solution

Step-by-Step Solution

Let's simplify the expression 53×24×52×23 5^3 \times 2^4 \times 5^2 \times 2^3 using the rules for exponents. We'll apply the product of powers rule, which states that when multiplying like bases, you can add the exponents.

  • Step 1: Focus on terms with the same base.
    Combine 53 5^3 and 52 5^2 . Since both terms have the base 55, we apply the rule am×an=am+na^m \times a^n = a^{m+n}: 53×52=53+2=55 5^3 \times 5^2 = 5^{3+2} = 5^5

  • Step 2: Combine 24 2^4 and 23 2^3 . Similarly, for the base 22: 24×23=24+3=27 2^4 \times 2^3 = 2^{4+3} = 2^7

After simplification, the expression becomes:
55×27 5^5 \times 2^7

Answer

55×27 5^5\times2^7

Exercise #14

Insert the corresponding expression:

6764= \frac{6^7}{6^4}=

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Identify the given information and relevant exponent rules.

  • Apply the quotient property of exponents.

  • Simplify the expression.

Now, let's work through each step:
Step 1: The problem gives us the expression 6764 \frac{6^7}{6^4} . The base is 6, and the exponents are 7 and 4, respectively.
Step 2: According to the rule of exponents, when dividing powers with the same base, we subtract the exponents: aman=amn \frac{a^m}{a^n} = a^{m-n} In this case, a=6 a = 6 , m=7 m = 7 , and n=4 n = 4 .
Step 3: Applying this rule gives us: 6764=674=63 \frac{6^7}{6^4} = 6^{7 - 4} = 6^3

Therefore, the solution to the problem is 63 6^3 .

Answer

63 6^3

Exercise #15

Simplify the following equation:

64×23×62×25= 6^4\times2^3\times6^2\times2^5=

Video Solution

Step-by-Step Solution

To simplify the equation 64×23×62×25 6^4 \times 2^3 \times 6^2 \times 2^5 , we will make use of the rules of exponents, specifically the product of powers rule, which states that when multiplying two powers that have the same base, you can add their exponents.

Step 1: Identify and group the terms with the same base.
In the expression 64×23×62×25 6^4 \times 2^3 \times 6^2 \times 2^5 , group the powers of 6 together and the powers of 2 together:

  • Powers of 6: 64×62 6^4 \times 6^2

  • Powers of 2: 23×25 2^3 \times 2^5

Step 2: Apply the product of powers rule.
According to the product of powers rule, for any real number a a , and integers m m and n n , the expression am×an=am+n a^m \times a^n = a^{m+n} .

Apply this rule to the powers of 6:
64×62=64+2=66 6^4 \times 6^2 = 6^{4+2} = 6^6 .

Apply this rule to the powers of 2:
23×25=23+5=28 2^3 \times 2^5 = 2^{3+5} = 2^8 .

Step 3: Write down the final expression.
Combining our results gives the simplified expression: 66×28 6^6 \times 2^8 .

Therefore, the solution to the problem is 66×28 6^6 \times 2^8 .

Answer

66×28 6^6\times2^8

Topics learned in later sections

  1. Rules of Exponentiation
  2. Combining Powers and Roots