5. Apply the Properties of Integer Exponents to generate equivalent expressions to this DIVISION problem
(6^-3)^7 x 6^20/6^-2
To generate equivalent expressions for the given division problem, we can apply the rules of exponents.
First, let's simplify the expression within the parentheses: (6^-3)^7 = 6^(-3*7) = 6^-21.
Now, we can substitute this simplified expression into the division problem: 6^-21 x 6^20/6^-2.
Next, we can apply the properties of exponents for multiplication and division.
For multiplication with the same base, we add the exponents: 6^-21 x 6^20 = 6^(-21+20) = 6^-1.
For division with the same base, we subtract the exponents: 6^-1/6^-2 = 6^(-1-(-2)) = 6^(-1+2) = 6^1 = 6.
Therefore, an equivalent expression for the given division problem is 6.
To generate equivalent expressions for this division problem using the properties of integer exponents, we can simplify the expression step by step:
Step 1: Start by simplifying within the parentheses.
(6^-3)^7 = 6^(-3 * 7) = 6^-21
Step 2: Now rewrite the entire expression using the simplified exponent.
6^-21 x 6^20/6^-2
Step 3: Use the property of exponents that states when dividing with the same base, subtract the exponents.
6^(-21 + 20) / 6^-2
Step 4: Simplify the exponent.
6^-1 / 6^-2
Step 5: Use the property of exponents that states when dividing with the same base, subtract the exponents again.
6^(-1 - (-2))
Step 6: Simplify the exponent once more.
6^(-1 + 2)
Step 7: Calculate the exponent.
6^1
Therefore, the final equivalent expression for the given division problem is 6.
To generate equivalent expressions for this division problem, we can use the properties of integer exponents. The properties we will utilize are:
1. Product Rule: a^m * a^n = a^(m+n)
2. Quotient Rule: a^m / a^n = a^(m-n)
3. Power Rule: (a^m)^n = a^(m*n)
4. Negative Exponent Rule: a^-n = 1/a^n
5. Zero Exponent Rule: a^0 = 1
Let's simplify the expression step by step:
Step 1: Simplify the powers of 6 within parentheses.
(6^-3)^7 x 6^20 / 6^-2
Applying the power rule, we multiply the exponents:
6^(-3*7) x 6^20 / 6^-2
Simplifying further using multiplication:
6^-21 x 6^20 / 6^-2
Step 2: Apply the product rule to multiply the bases with the same exponents.
6^(-21 + 20) / 6^-2
Simplifying inside parentheses:
6^-1 / 6^-2
Step 3: Apply the quotient rule by subtracting the exponents.
6^(-1 - (-2))
Simplifying the subtraction:
6^(1)
Therefore, the equivalent expression to the given division problem is 6^1.