Understanding Exponents and Inverses
Exponents are shorthand for repeated multiplication. For instance, 2^3 means 2 multiplied by itself three times, or 2 \times 2 \times 2 = 8. The inverse operation of exponentiation is finding the logarithm. However, when dealing with fractions as bases and negative exponents, simplifying expressions can become complex.
The expression \left(\frac{1}{7}\right)^{-7} involves a fraction raised to a negative exponent. To simplify this, we need to apply the rules of exponents and fractions.
Rule 1: Negative Exponents
A negative exponent indicates taking the reciprocal of the base to the power of the absolute value of the exponent. Therefore, \left(\frac{1}{7}\right)^{-7} can be rewritten as \left(\frac{7}{1}\right)^7.
📝 Note: When converting a negative exponent to a positive one, we flip the fraction and make the exponent positive.
Rule 2: Exponentiation of a Fraction
When a fraction is raised to a power, both the numerator and the denominator are raised to that power. Thus, \left(\frac{7}{1}\right)^7 becomes \frac{7^7}{1^7}.
📝 Note: The denominator, $1$, raised to any power remains $1$.
Rule 3: Simplifying the Expression
Since the denominator is 1 and any number raised to the power of 0 is 1, the expression simplifies to 7^7. This is because 1^7 = 1, making the denominator irrelevant.
Calculating the Value of $7^7$
To calculate 7^7, we simply multiply 7 by itself seven times: 7 \times 7 \times 7 \times 7 \times 7 \times 7 \times 7.
Using Exponent Rules for Faster Computation
For expressions like 7^7, where the exponent is large, we can use the rule of exponentiation that states (a^m)^n = a^{m \cdot n}. However, since we’re not dealing with nested exponents here, we’ll focus on direct computation.
Direct Computation of $7^7$
Calculating 7^7 directly gives us 7^7 = 823,543.
Conclusion
Simplifying the inverse of \left(\frac{1}{7}\right)^{-7} involves understanding the rules of negative exponents, exponentiation of fractions, and simplifying expressions. By applying these rules, we find that \left(\frac{1}{7}\right)^{-7} = 7^7 = 823,543.
What is the rule for negative exponents?
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A negative exponent indicates taking the reciprocal of the base to the power of the absolute value of the exponent.
How do you simplify an expression with a fraction raised to a power?
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Both the numerator and the denominator are raised to that power. The denominator remains 1 if it’s raised to any power, simplifying the fraction to the numerator raised to that power.
What is the final value of 7^7?
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7^7 = 823,543.
