Problem 78
Question
Simplify each expression. $$ x^{2} x^{15} x^{9} $$
Step-by-Step Solution
Verified Answer
The expression simplifies to \(x^{26}\).
1Step 1: Understand the Question
The given problem is to simplify the expression \(x^2 \times x^{15} \times x^9\). This means we need to combine the powers of \(x\) into a single term by applying the laws of exponents.
2Step 2: Apply the Product of Powers Property
According to the product of powers property, when multiplying like bases you add their exponents. Hence, we add the exponents: 2, 15, and 9. \[x^{2} \times x^{15} \times x^{9} = x^{(2+15+9)}\]
3Step 3: Perform the Addition
Add the exponents together. Calculating the sum:\[2 + 15 + 9 = 26\]
4Step 4: Write the Simplified Expression
Substitute the sum of the exponents back into the expression:\[x^{26}\]The expression \(x^2 \times x^{15} \times x^9\) simplifies to \(x^{26}\).
Key Concepts
Product of Powers PropertySimplifying ExpressionsExponents Addition
Product of Powers Property
To understand how to simplify expressions involving exponents, it's essential to grasp the **Product of Powers Property**. This property is a fundamental rule in exponentiation that states: when you multiply two expressions with the same base, you can add their exponents. The base remains unchanged, and only the exponents are added together.
In mathematical terms, if you have expressions like \[a^m \times a^n\] where \(a\) is the base and \(m\) and \(n\) are the exponents, the result is \[a^{m+n}\].
Applying this property makes it easier to handle expressions without doing individual multiplication for potentially huge numbers. It's a time-saver and reduces the complexity significantly.
In mathematical terms, if you have expressions like \[a^m \times a^n\] where \(a\) is the base and \(m\) and \(n\) are the exponents, the result is \[a^{m+n}\].
Applying this property makes it easier to handle expressions without doing individual multiplication for potentially huge numbers. It's a time-saver and reduces the complexity significantly.
Simplifying Expressions
Simplifying expressions is about rewriting them in their simplest form. With exponents, simplifying is often about using the **Laws of Exponents** to combine terms where possible. This can make equations easier to work with, as they're presented in a more compact form.
Consider the expression \[x^2 \times x^{15} \times x^9\]. Each term has a common base \(x\). By recognizing this, you can apply the laws of exponents to combine them. The whole process results in a single term with a new, simplified exponent, which reduces the clutter in an equation.
Learning how to simplify such expressions equips you with tools to solve more complex algebraic problems because it breaks down intricate expressions into manageable pieces that reveal the essence of the problem.
Consider the expression \[x^2 \times x^{15} \times x^9\]. Each term has a common base \(x\). By recognizing this, you can apply the laws of exponents to combine them. The whole process results in a single term with a new, simplified exponent, which reduces the clutter in an equation.
Learning how to simplify such expressions equips you with tools to solve more complex algebraic problems because it breaks down intricate expressions into manageable pieces that reveal the essence of the problem.
Exponents Addition
When combining terms with the same base, the main operation performed is adding the exponents. This **Exponents Addition** comes directly from the Product of Powers Property, and it's a key technique in algebra.
In our example, to simplify \[x^2 \times x^{15} \times x^9\], you perform the exponent addition:
By mastering exponent addition, you gain skills to effectively streamline and manage algebraic expressions, making further operations clearer and less error-prone.
In our example, to simplify \[x^2 \times x^{15} \times x^9\], you perform the exponent addition:
- Identify the exponents: 2, 15, and 9.
- Add these exponents together: \[2 + 15 + 9 = 26\].
By mastering exponent addition, you gain skills to effectively streamline and manage algebraic expressions, making further operations clearer and less error-prone.
Other exercises in this chapter
Problem 78
Write each number in scientific notation. $$ 0.00194 $$
View solution Problem 78
Mixed Practice Multiply. $$ (4 x-9 y)^{2} $$
View solution Problem 79
Fill in the squares so that each is a true statement. $$ 3 x^{\square}+4 x^{2}=7 x^{\square} $$
View solution Problem 79
Simplify, if possible. $$ a \cdot b^{3} \cdot a^{2} \cdot b^{7} $$
View solution