91Ó°ÊÓ

Positive exponents tell you how many times to multiply a base number by itself. For example, in the expression \(5^3\), the base is 5 and the exponent is 3. This means you multiply 5 by itself three times: \(5^3 = 5 \cdot 5 \cdot 5 = 125\).
Exponents are a way to simplify multiplication, making problems easier to solve and understand.

• Base: The base is the number that is being multiplied.
• Exponent: The exponent shows how many times the base is used as a factor.

Remember, even exponents always result in a positive number. For instance, \((-2)^6 = 64\) since multiplying a negative number an even number of times gives a positive result.

The reciprocal of a number is another number which, when multiplied together, equals 1. For a fraction, the reciprocal is obtained by swapping the numerator and the denominator. For instance, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).
Reciprocals are especially important when dealing with negative exponents. A negative exponent indicates that you should take the reciprocal of the base and then apply the positive exponent. For example, \(a^{-n} = \frac{1}{a^n}\). In the exercise, \((-2)^{-6}\) is rewritten as \(\frac{1}{(-2)^6}\), making it much easier to solve.

• Negative exponents use reciprocals to simplify expressions.

Simplifying expressions involves reducing them to their most basic form. This can make calculations easier and results clearer. When an expression involves exponents, taking the following steps can help simplify it:

1. Rewrite any terms with negative exponents as their reciprocals with positive exponents.
2. Evaluate the positive exponents, simplifying the multiplications.
For example, in the given exercise \((-2)^{-6}\), we first convert to a positive exponent by finding the reciprocal, giving us \(\frac{1}{(-2)^6}\). Then we calculate the positive exponent: \((-2)^6 = 64\). Finally, simplify the fraction to get \(\frac{1}{64}\).

• Always start by converting negative exponents.
• Calculate any remaining exponents for simplification.
• Always reduce fractions if possible.

Simplifying ensures your results are accurate and easy to understand.