Question Video: Simplifying Algebraic Expressions Using Laws of Exponents | Nagwa Question Video: Simplifying Algebraic Expressions Using Laws of Exponents | Nagwa

Question Video: Simplifying Algebraic Expressions Using Laws of Exponents Mathematics • First Year of Preparatory School

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Simplify 𝑏⁵ × 𝑏².

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Video Transcript

Simplify 𝑏 raised to the fifth power times 𝑏 squared.

In this question, we are asked to simplify a product. We can note that each factor is an exponential expression with the same base. We can then recall that we can simplify the product of exponential expressions by using the product rule for exponents, which tells us that 𝑏 raised to the power of 𝑚 times 𝑏 raised to the power of 𝑛 is equal to 𝑏 raised to the power of 𝑚 plus 𝑛, for any nonnegative integers 𝑚 and 𝑛.

In other words, we can find the product of these exponential expressions with the same base by raising the base 𝑏 to the sum of the exponents. We obtain 𝑏 raised to the power of five plus two. We can then evaluate the expression in the exponent to obtain 𝑏 raised to the seventh power.

While this is enough to answer the question, it can be useful to show this calculation in full to show why this result is actually true. We can do this by recalling that raising a number to a positive integer exponent 𝑛 is the same as multiplying 𝑛 lots of that number together. Therefore, 𝑏 raised to the power of five is 𝑏 times 𝑏 times 𝑏 times 𝑏 times 𝑏, and 𝑏 squared is 𝑏 times 𝑏. We see that the product of these expressions can be thought of as the product of five plus two lots of 𝑏.

Therefore, we can write this product as 𝑏 raised to the power of five plus two, which is equal to 𝑏 raised to the seventh power. This outlines the proof of the product rule for exponents with positive integer exponents. However, we have shown that 𝑏 raised to the fifth power times 𝑏 squared is equal to 𝑏 raised to the seventh power.

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