A student incorrectly claims that since 2 x^2 ·2 x^2=4 x^4, it follows that 5 x^5 ·5 x^5=25 x^25

step 1 A student incorrectly claims that since 2x square into 2x square is equal to 4x to the power 4,

A student incorrectly claims that since 2 x^2 ·2 x^2=4 x^4,...

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Key Concepts

Multiplying Coefficients Separately from Variables

In algebra, it is important to distinguish between the numerical coefficients and the variable parts of an expression. The coefficients are multiplied together independently of the variables. In the provided example, multiplying 5 by 5 yields 25. Afterward, the exponent rule for the variables is applied separately, which clarifies that the initial mistake came from misapplying exponent rules rather than an error with the coefficients.

Common Misconceptions in Exponent Operations

A frequent error among students involves mistaking the rules of multiplication for exponents. Instead of adding exponents when multiplying powers of the same base, one may incorrectly multiply the exponents, as was seen in the mistaken claim. Clarifying that exponents represent repeated multiplication and how the exponent laws operate helps prevent such misconceptions and underlines the proper method for handling exponents.

Exponent Addition Rule

When multiplying expressions with the same base, such as x^a and x^b, the correct rule is to add the exponents (x^(a+b)) rather than multiply them. This fundamental property of exponents ensures that the growth of the exponent is linear, not exponential, during multiplication, so in the example, 5x^5 multiplied by 5x^5 becomes 25x^(5+5), or 25x^10.

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