Algebra

7^8 · 7^-9

Step-by-step solution with explanation

Final Answer

Step-by-step solution

1

Identify the bases and exponents

Both terms have the same base of 7. One exponent is 8 and the other is -9. When bases match, we can use the product rule for exponents.
2

Apply the product rule for exponents

The product rule says: when you multiply same bases, add the exponents. So we add 8 and -9 together.
3

Add the exponents together

Adding a negative is the same as subtracting. 8 minus 9 equals -1, so our new exponent is -1.
4

Write the simplified result

We now have 7 raised to the power of -1. A negative exponent means we take the reciprocal of the base.
5

Convert negative exponent to a fraction

Any number raised to -1 equals 1 divided by that number. So .

Understanding this problem

Learning Insight

The product rule works because exponents count how many times you multiply a number by itself. Adding exponents combines those counts into one. A negative exponent doesn't mean a negative answer — it means you flip the base into a fraction (take the reciprocal).

Quick Tip

Same base, multiplying? Just ADD the exponents. Remember: negative exponent = flip it into a fraction. So .

Common Mistake

Students often multiply the exponents instead of adding them, getting instead of . Multiplying exponents is the rule for a POWER of a power like , not for multiplying two separate terms.