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Expressions with are mathematical terms that involve variables raised to exponents. These types of can be intimidating at first, but with a few key strategies, they can become much more manageable. In this article, we will explore the steps to solving s with powers, providing you with a solid foundation to tackle this concept.
Step 1: Understanding the Basics
Before diving into solving expressions with powers, it is important to familiarize yourself with the basics. Powers, also known as exponents, are used to express repeated multiplication of the same number. For instance, in the expression 2³, the base number is 2, and the exponent, or power, is 3. This means that we have three instances of multiplying 2 by itself: 2 × 2 × 2, resulting in 8. Keep in mind that the exponent indicates the number of times the base is multiplied by itself.
Step 2: Simplify the Expression
The first step to solving expressions with powers is to simplify them. Simplifying an expression involves performing any necessary arithmetic operations to reduce it to its simplest form. For example, take the expression 5² × 5³. We can simplify this by first calculating 5², which equals 25, and 5³, which equals 125. Then, we multiply these results together: 25 × 125, resulting in 3,125. So, 5² × 5³ simplifies to 3,125.
Step 3: Applying the Laws of Exponents
Once you have simplified the expression, you can apply the laws of exponents to it further. The laws of exponents provide rules for manipulating expressions with powers. Here are some fundamental laws to remember:
1. Product of Powers:
When multiplying two powers with the same base, you can add the exponents. For instance, if you have x² × x³, you can combine the exponents to get x⁵.
2. Quotient of Powers:
When dividing two powers with the same base, you can subtract the exponents. For example, if you have x⁷ ÷ x³, you can subtract the exponents to get x⁴.
3. Power of Power:
When raising a power to another power, you can multiply the exponents. For instance, if you have (x²)³, you can multiply the exponents to get x⁶.
4. Negative Exponent:
A negative exponent indicates that the base should be placed in the denominator. For example, x⁻³ is equivalent to 1/x³.
Step 4: Simplify Further, if Possible
After applying the laws of exponents, you should aim to simplify the expression even more, if possible. Combining like terms and reducing fractions are common techniques used to achieve further simplification.
Step 5: Follow Order of Operations
Lastly, it is essential to follow the order of operations when solving expressions with powers. This involves considering any parentheses, exponents, multiplication, division, addition, and subtraction in the correct sequence to obtain the accurate result.
By following these steps, you can approach expressions with powers with confidence. Remember to simplify the expression, apply the laws of exponents, and simplify further, if possible. Always adhere to the order of operations to ensure an accurate solution. With practice, solving expressions with powers will become second nature, and you will be equipped to tackle more complex mathematical problems.
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