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Chapter 10: Problem 524

Simplify each expression. $$p^{3} \cdot p^{10}$$

Short Answer

Expert verified
The simplified expression is \( p^{13} \).

Step by step solution

01

Identify the Base and the Exponents

In the expression, the base is the variable 'p'. The exponents are 3 and 10.
02

Apply the Product of Powers Property

The Product of Powers Property states that when multiplying two exponents with the same base, you can add the exponents. For this expression, it means: \( p^{3} \times p^{10} = p^{(3+10)} \)
03

Simplify the Exponent

Add the exponents together: \( p^{(3+10)} = p^{13} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product of Powers
In algebra, you will often come across expressions where you need to multiply exponents. This is where the 'Product of Powers' property comes in handy. The rule is simple: when you multiply two expressions that have the same base, you add their exponents.

Let's consider the example given: \( p^{3} \times p^{10} \).
Here, the base is 'p', and the exponents are 3 and 10. Instead of multiplying the exponents, you add them: \( p^{3} \times p^{10} = p^{3+10} \).
Simplifying further, you get: \( p^{13} \). This makes solving expressions much quicker and easier!
Exponent Addition
Adding exponents becomes straightforward once you understand the Product of Powers property. By recognizing that you can add the exponents when the bases are the same, solving these types of expressions becomes a breeze.

In our example, we had \( p^{3} \times p^{10} \). The bases are both 'p', so we simply add the exponents 3 and 10: \( 3 + 10 = 13 \). Thus, \( p^{(3+10)} = p^{13} \).

Remember, this rule applies only when the base remains the same. Different bases can't be combined in this way.
Simplifying Expressions
Simplifying expressions is a fundamental skill in algebra. To simplify means to reduce the expression to its most basic form.

Let's look back at the problem: \( p^{3} \times p^{10} \). Using the Product of Powers property, we combined the exponents: \( 3 + 10 \). This gave us \( p^{13} \).

When you simplify, you make the expression cleaner and easier to work with. Always look for opportunities to combine like terms or use properties like the Product of Powers to achieve this. Regular practice will make these steps feel almost automatic.

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Most popular questions from this chapter

A hiker drops a pebble from a bridge 240 feet above a canyon. The polynomial \(-16 t^{2}+240\) gives the height of the pebble \(t\) seconds a after it was dropped. Find the height when \(t=3\) .

Multiply and write your answer in decimal form. $$\left(3 \times 10^{7}\right)\left(2 \times 10^{-4}\right)$$

Simplify each expression. $$p^{-3} \cdot p^{-8}$$

Simplify. $$\frac{\left(n^{5}\right)^{3}}{\left(n^{2}\right)^{8}}$$

Simplify. $$(8 n)^{-1}$$
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