Find the Exact Value ((2^-10)/(4^2))^7
The question asks you to calculate the exact value of a complex fraction raised to a power. The fraction involves negative exponents and powers, specifically the negative power of 2 (2^-10) divided by a positive power of 4 (4^2), and this entire fraction is then raised to the seventh power. The process will involve rules of exponents and simplification of the expression to its smallest form without using a calculator.
$\left(\left(\right. \frac{2^{- 10}}{4^{2}} \left.\right)\right)^{7}$
Answer
Solution:
Step 1:
Apply the negative exponent rule $a^{-n} = \frac{1}{a^n}$ to transform $2^{-10}$ to the denominator.
$$\left(\frac{2^{-10}}{4^2}\right)^7 = \left(\frac{1}{2^{10} \cdot 4^2}\right)^7$$
Step 2:
Begin simplifying the expression in the denominator.
Step 2.1:
Express $4$ as $2^2$.
$$\left(\frac{1}{(2^2)^2 \cdot 2^{10}}\right)^7$$
Step 2.2:
Handle the exponent on the exponent in $(2^2)^2$.
Step 2.2.1:
Utilize the power of a power rule, $(a^m)^n = a^{mn}$.
$$\left(\frac{1}{2^{2 \cdot 2} \cdot 2^{10}}\right)^7$$
Step 2.2.2:
Calculate $2 \cdot 2$.
$$\left(\frac{1}{2^4 \cdot 2^{10}}\right)^7$$
Step 2.3:
Combine exponents on the same base using $a^m \cdot a^n = a^{m+n}$.
$$\left(\frac{1}{2^{4+10}}\right)^7$$
Step 2.4:
Add the exponents $4$ and $10$.
$$\left(\frac{1}{2^{14}}\right)^7$$
Step 3:
Calculate $2^{14}$.
$$\left(\frac{1}{16384}\right)^7$$
Step 4:
Raise the fraction to the seventh power using the power of a quotient rule, $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.
$$\frac{1^7}{16384^7}$$
Step 5:
Recognize that any number to the power of one remains unchanged.
$$\frac{1}{16384^7}$$
Step 6:
Present the result in its exact and decimal forms.
Exact Form: $$\frac{1}{16384^7}$$ Decimal Form: $$3.15544362 \cdot 10^{-30}$$
Knowledge Notes:
Negative Exponent Rule: The negative exponent rule states that $a^{-n} = \frac{1}{a^n}$. This rule is used to transform negative exponents into positive exponents by moving the base to the opposite side of the fraction.
Power of a Power Rule: The power of a power rule, $(a^m)^n = a^{mn}$, is used when an exponent is raised to another exponent. The rule states that you multiply the exponents together.
Power of a Quotient Rule: The power of a quotient rule, $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$, is used when a fraction is raised to an exponent. The rule states that you apply the exponent to both the numerator and the denominator separately.
Combining Exponents Rule: When you have the same base being multiplied with different exponents, you can combine the exponents by adding them together, $a^m \cdot a^n = a^{m+n}$.
Simplifying Exponents: When simplifying expressions with exponents, it is often helpful to rewrite numbers in terms of a common base to make the application of exponent rules easier.
Exact vs Decimal Form: The exact form of an expression is the form that uses integers and exponents without approximation. The decimal form is a numerical approximation of the exact form, often rounded to a certain number of significant figures.
