Exponents and the Laws of Exponents for Real Numbers

Exponents are a mathematical notation used to represent repeated multiplication. They are a way to express numbers raised to a certain power or exponent. The Laws of Exponents are a set of rules that govern the manipulation and simplification of expressions with exponents.

Exponentiation

Exponentiation is the process of raising a number to a certain power. It is represented using the "^" symbol. For example, 2^3 means 2 raised to the power of 3, which equals 2 x 2 x 2 = 8.

Multiplication of Exponents

When multiplying two numbers with the same base but different exponents, we can use the Law of Exponents to simplify the expression. According to this law, when multiplying two numbers with the same base, we add their exponents. For example, 2^3 x 2^2 = 2^(3+2) = 2^5 = 32.

Calculating (625)0.16

To calculate (625)0.16, we need to find the value of 625 raised to the power of 0.16. We can rewrite 0.16 as a fraction, where the numerator is 16 and the denominator is 100. This is because raising a number to a fractional exponent is equivalent to taking the corresponding root of the number.

Therefore, (625)0.16 = 625^(16/100).

Simplifying the Expression

To simplify the expression further, we can rewrite 625 as 5^4. Therefore, (625)0.16 becomes (5^4)^(16/100).

According to the Law of Exponents, when raising a power to another power, we multiply the exponents. Applying this rule, we have (5^4)^(16/100) = 5^(4*(16/100)) = 5^(64/100).

Calculating 5^(64/100)

To calculate 5^(64/100), we can rewrite 64/100 as a decimal. Dividing 64 by 100 gives us 0.64.

Therefore, 5^(64/100) = 5^0.64.

Calculating (625)0.9

Similarly, to calculate (625)0.9, we need to find the value of 625 raised to the power of 0.9. Using the same approach as before, we have (625)0.9 = 625^(9/10).

Simplifying the Expression

We can rewrite 625 as 5^4. Therefore, (625)0.9 becomes (5^4)^(9/10).

Applying the Law of Exponents, (5^4)^(9/10) = 5^(4*(9/10)) = 5^(36/10).

Calculating 5^(36/10)

To calculate 5^(36/10), we can simplify 36/10 as 3.6.

Therefore, 5^(36/10) = 5^3.6.

Conclusion

In conclusion, to calculate (625)0.16 and (625)0.9,

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