Apply the product rule of integer exponents to find the numerical equivalent of 12−5⋅127 . Show your work and explain how you used the product rule of integer exponents.
To apply the product rule of integer exponents, we need to multiply the base numbers and then apply the exponent to the product.
Here, the base number is 12 and the exponent is -5.
First, let's rewrite the expression using the product rule of integer exponents:
12^(-5) * 127
According to the product rule, we multiply the base numbers, so let's calculate 12^(-5) first:
12^(-5) = 1 / 12^(5)
Now, we can substitute this value back into our expression:
(1 / 12^(5)) * 127
To find the numerical equivalent, we need to evaluate this expression:
1 / 12^(5) = 1 / (12 * 12 * 12 * 12 * 12) = 1 / 248,832
Finally, we multiply the result by 127:
(1 / 248,832) * 127 = 0.000508 [rounded to 6 decimal places]
Therefore, 12^(-5) * 127 is approximately equal to 0.000508.
To apply the product rule of integer exponents, we use the formula:
a^m ⋅ a^n = a^(m+n)
In this case, we have the expression 12^(-5) ⋅ 127.
Using the product rule of integer exponents, we can rewrite this expression as:
12^(-5) ⋅ 127 = (12 ⋅ 127)^(−5)
To simplify further, we can calculate the value of 12 ⋅ 127:
12 ⋅ 127 = 1524
Now, we substitute this value back into our expression:
(12 ⋅ 127)^(−5) = 1524^(-5)
To evaluate 1524^(-5), we use the rule that a number raised to the power of a negative exponent is equal to 1 divided by that number raised to the power of the positive exponent:
1524^(-5) = 1 / (1524^5)
To find the numerical equivalent, we can calculate this value using a calculator or mathematical software.
It is important to note that we applied the product rule of integer exponents to combine the exponents of 12 and 127 before evaluating the expression.
To apply the product rule of integer exponents, we need to remember the rule: When you multiply two numbers with the same base but different exponents, you add their exponents.
In this case, we have 12^(-5) ⋅ 12^7.
To find the numerical equivalent, we can simplify each term separately and then multiply the results.
First, let's simplify 12^(-5):
12^(-5) = 1 / 12^5
Next, let's simplify 12^7:
12^7 = 12 ⋅ 12 ⋅ 12 ⋅ 12 ⋅ 12 ⋅ 12 ⋅ 12 = 358,318,080
Now, we can multiply the simplified terms:
1 / 12^5 ⋅ 358,318,080 = 1 / (12^5) ⋅ 358,318,080 = 1 / 248,832 ⋅ 358,318,080
To divide 1 by 248,832, we can rewrite 1 as a fraction with 248,832 as the denominator:
1 / 248,832 = 1 ÷ 248,832 = 0.0000040198 (approx)
Finally, we multiply the result by 358,318,080:
0.0000040198 ⋅ 358,318,080 ≈ 1443.1016447
Therefore, the numerical equivalent of 12^(-5) ⋅ 12^7 is approximately 1443.1016447.
To summarize, we used the product rule of integer exponents by multiplying each term separately, then simplifying them, and finally multiplying the results.