Question

How are the expressions \(x^{7} \cdot x^{3}\) and \(\left(x^{7}\right)^{3}\) different? Explain your answer.

Short answer

The expressions \(x^{7} \cdot x^{3}\) and \(\left(x^{7}\right)^{3}\) are different because they follow different laws of exponents. The first expression results in \(x^{10}\) and the second results in \(x^{21}\) due to the rules of adding and multiplying exponents, respectively.

Step-by-step solution

Apply the law of exponents to \(x^{7} \cdot x^{3}\)

When multiplying two exponential terms with the same base, add the exponents. In this case, the base is \(x\) and the exponents are 7 and 3. Adding these exponents gives \(x^{7+3} = x^{10}\).

Apply the law of exponents to \(\left(x^{7}\right)^{3}\)

When raising an exponential term to a power, multiply the exponents. In this case, the base is \(x\) and the first exponent is 7, which is then raised to the 3rd power. Multiplying these exponents gives \(\left(x^{7}\right)^{3} = x^{7*3} = x^{21}\).

Compare the Results

The result of the first operation, \(x^{7} \cdot x^{3}\), is \(x^{10}\), and the result of the second operation, \(\left(x^{7}\right)^{3}\), is \(x^{21}\). Clearly, these are not the same, demonstrating how the expressions are different.

Key concepts

Exponential Expressions

Exponential expressions are a core concept in understanding how to work with powers in mathematics. They are written in the form of a base number raised to an exponent, represented as \( b^n \), where \( b \) is the base and \( n \) is the exponent, indicating how many times the base is multiplied by itself.

For example, the expression \( 3^4 \) signifies that the number 3 is multiplied by itself 4 times: \( 3 \times 3 \times 3 \times 3 = 81 \). The base, \( 3 \), is the consistent factor, and the exponent, \( 4 \), tells us the number of occurrences in the multiplication. Simplifying exponential expressions requires a good grasp of the properties of exponents, which includes operations like multiplying exponents, as well as raising exponents to a power.

Multiplying Exponents

When it comes to multiplying exponents with the same base, the process is straightforward: you simply add the exponents together. This is known as the Product of Powers Property, and it's used to simplify expressions where two exponential terms are multiplied.

Let's apply this rule to the example given: \( x^7 \cdot x^3 \). Since both terms have the same base of \( x \), you can add the exponents together to get \( x^{7+3} \) resulting in \( x^{10} \).

This fundamental principle allows us to combine powers efficiently and is crucial for solving more complex algebraic problems involving exponents. Remember that this property only applies when the bases are the same; if they differ, this method isn't applicable.

Raising Exponent to a Power

When you raise an exponent to a power, you're essentially dealing with 'a power to a power.' This situation calls for multiplying the exponents together, which is due to the Power of a Power Property. It allows you to condense the expression without altering its value.

Taking the exercise's example \( (x^7)^3 \), you'll multiply the exponents: 7 (from \( x^7 \) ) and 3 (the outer exponent). This multiplication \( 7 \times 3 \) results in \( x^{21} \).

It's important to note that this operation intensifies the expression, as the base is raised to a much higher power than in the multiplication scenario. Distinguishing when to add or multiply exponents is key in manipulating and simplifying expressions with powers effectively.