Question

Simplify: \(e^{2} \cdot e^{5}=\) ________ ;\(\left(e^{4}\right)^{3}=\) __________.

Short answer

e^{7}, e^{12}

Step-by-step solution

Apply the Product Rule for Exponents

The product rule for exponents states that when multiplying two exponents with the same base, you can add the exponents. For the expression \(e^{2} \times e^{5}\), add the exponents: \(2 + 5 = 7\).Thus,\(e^{2} \times e^{5} = e^{7}\).

Apply the Power Rule for Exponents

The power rule for exponents states that when raising an exponent to another exponent, you multiply the exponents. For the expression \( (e^{4})^{3} \), multiply the exponents: \(4 \times 3 = 12\).Thus, \((e^{4})^{3} = e^{12}\).

Key concepts

Product Rule for Exponents

The product rule for exponents is a fundamental concept in algebra that helps to simplify expressions involving exponents with the same base. According to this rule, when you multiply two exponents that have the same base, you can add their exponents together.

For example, in the expression e^{2} \times e^{5}, both exponents share the same base e. To simplify this expression, you add the exponents:

\[e^{2} \times e^{5} = e^{2+5} = e^{7}\]

This rule is incredibly useful when dealing with more complex exponential expressions, as it allows you quickly to reduce them to simpler forms. Always remember: only apply this rule when the bases are the same.

Power Rule for Exponents

The power rule for exponents is another vital principle in algebra. This rule states that when you raise an exponent to another exponent, you multiply the exponents.

For example, if you have the expression (e^{4})^{3}, you raise the base e with an exponent of 4 to the power of 3. According to the power rule, you multiply the exponents together:

\[(e^{4})^{3} = e^{4 \times 3} = e^{12}\]

This rule helps to simplify expressions where exponents are nested within other exponents, making it easier to manage and solve complex mathematical problems.

Simplifying Exponents

Simplifying exponents involves using rules such as the product and power rule to make complex expressions more manageable. Combining these rules allows you to rewrite exponents in ways that are easier to understand and work with.

For instance, we have the exercise:

• Simplify e^{2} \times e^{5}; Applying the product rule: \[e^{2} \times e^{5} = e^{7}\]
• Simplify (e^{4})^{3}; Applying the power rule: \[(e^{4})^{3} = e^{12}\]

By consistently using these rules, you can simplify a wide range of exponential expressions. Always take your time to identify which rule applies to the situation at hand to ensure correct simplification.