Apply the Properties of Integer Exponents to generate equivalent expressions to this DIVISION problem
(6^-3)^7 x 6^20
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6^-2
To generate equivalent expressions for this division problem using the properties of integer exponents, we can apply the following rules:
1. When dividing two terms with the same base, subtract the exponents.
2. When raising a power to another power, multiply the exponents.
3. Any number raised to the power of 0 is equal to 1.
Applying these rules, we can rewrite the expression as follows:
(6^-3)^7 x 6^20 ÷ 6^-2
First, we can simplify the division part by subtracting the exponents:
(6^-3)^7 x 6^20 ÷ 6^-2 = (6^-3)^7 x 6^(20-(-2))
Next, we can simplify the subtraction:
(6^-3)^7 x 6^(20+2)
Now, we can simplify the expression inside the parentheses:
(6^-3)^7 x 6^22
Since we have the same base (6), we can multiply the exponents:
6^(-3 x 7) x 6^22
Simplifying further:
6^-21 x 6^22
Now, we can apply the rule that any number raised to the power of 0 is equal to 1:
6^(-21 + 22)
6^1
Therefore, the equivalent expression to the division problem is simply 6.
To generate equivalent expressions for the given division problem, we can use the properties of integer exponents:
Property 1: (a^m)^n = a^(m * n)
Property 2: a^m * a^n = a^(m + n)
Property 3: a^m / a^n = a^(m - n)
Let's break down the problem step-by-step:
Step 1: Simplify the expression (6^-3)^7:
Using Property 1, we have (6^-3)^7 = 6^(-3 * 7) = 6^-21
Step 2: Simplify the expression 6^20:
This expression is already simplified and cannot be further simplified.
Step 3: Rewrite the expression:
Now, we can rewrite the division problem (6^-3)^7 * 6^20 ÷ 6^-2 using the equivalent expressions we found in Steps 1 and 2:
= 6^-21 * 6^20 ÷ 6^-2
Step 4: Simplify the expression using Property 2:
Applying Property 2, we can simplify the expression to:
= 6^(-21 + 20) ÷ 6^-2
= 6^-1 ÷ 6^-2
Step 5: Simplify the expression using Property 3:
Finally, applying Property 3, we can simplify the expression to:
= 6^(-1 - (-2))
= 6^1
= 6
Therefore, the equivalent expression to the given division problem is 6.
To generate equivalent expressions for this division problem: ((6^-3)^7 x 6^20) / 6^-2 using the properties of integer exponents, we can simplify the exponents and rewrite the expression.
First, let's simplify the exponents within the parentheses:
(6^-3)^7 = 6^(-3 x 7) = 6^-21
Next, we can rewrite the entire expression using the properties of exponents:
((6^-3)^7 x 6^20) / 6^-2 = (6^-21 x 6^20) / 6^-2
Now, let's apply the property of exponents that states: a^m / a^n = a^(m-n)
In this case, 6^-21 / 6^-2 is equal to 6^(-21 - (-2)) = 6^(-21 + 2) = 6^-19.
Therefore, the final equivalent expression is: 6^-19.