Simplifying Algebraic Expressions: A Step-by-Step Guide
Hey guys! Today, we're diving into the world of simplifying algebraic expressions. It might sound intimidating, but trust me, it's like solving a puzzle. We'll break down each expression step by step, so you can easily follow along and master these skills. Let's get started!
(a) Simplifying 10³ × 33 × 72
When we're faced with an expression like 10³ × 33 × 72, our goal is to break it down into its simplest form by performing the indicated operations. The key here is understanding exponents and basic arithmetic. Exponents tell us how many times a number is multiplied by itself. In this case, 10³ means 10 × 10 × 10, which equals 1000. So, the first step is to calculate 10³. Once we know that 10³ = 1000, we can substitute it back into the original expression, giving us 1000 × 33 × 72. Now, it’s just a matter of multiplying these numbers together. You can grab a calculator or do it manually; either way, the order of multiplication doesn't matter because multiplication is associative. That means (a × b) × c is the same as a × (b × c). Let's start by multiplying 1000 × 33, which gives us 33000. Now we have 33000 × 72. Multiplying these two numbers results in 2,376,000. So, the simplified form of 10³ × 33 × 72 is 2,376,000. Remember, always double-check your calculations to ensure accuracy. Simplification is all about breaking down complex expressions into manageable steps and then meticulously performing each step to arrive at the final answer. This approach not only helps in solving mathematical problems but also enhances your problem-solving skills in general. Practice makes perfect, so keep at it, and you’ll become a pro at simplifying expressions in no time!
(b) Simplifying x × 49 × 73 × 100
Now, let's tackle the expression x × 49 × 73 × 100. In this case, we have a variable, x, which means we can't get a single numerical answer, but we can still simplify the expression by combining the constants. Start by multiplying the numbers together: 49 × 73 × 100. It’s often easiest to start with the multiplication involving 100 since multiplying by 100 just means adding two zeros to the end of the number. So, let's multiply 49 × 73 first. 49 × 73 equals 3577. Now, multiply this result by 100: 3577 × 100 = 357,700. Now we have simplified the numerical part of the expression. The expression becomes x × 357,700. To write this in its simplest form, we just put the number in front of the variable: 357,700x. This is the simplified form of the expression. When simplifying expressions with variables, always remember to combine the constants first. This makes the expression cleaner and easier to work with. Understanding how to manipulate variables and constants is a fundamental skill in algebra, and it's crucial for solving more complex equations and problems. Keep practicing these simplification techniques, and you'll find that algebraic manipulations become second nature. Always double-check your calculations and make sure you’re comfortable with each step. With a bit of practice, you’ll be simplifying expressions like a math whiz in no time!
(c) Simplifying a³ × b² × b³
Let's move on to the expression a³ × b² × b³. This involves variables with exponents, so we need to use the rules of exponents to simplify it. Specifically, when multiplying terms with the same base, you add their exponents. In this case, we have a³ and b² × b³. Notice that a³ is already in its simplest form with respect to a, but we can simplify b² × b³. Since both terms have the same base (b), we can add their exponents: 2 + 3 = 5. So, b² × b³ = b⁵. Now, we can rewrite the original expression as a³ × b⁵. There are no more like terms to combine, so this is the simplest form of the expression. The simplified expression is a³b⁵. Remember, the key to simplifying expressions with exponents is to identify terms with the same base and then apply the appropriate exponent rules. This not only simplifies the expression but also makes it easier to understand and work with in further calculations. Keep practicing these rules, and you'll become more comfortable manipulating expressions with exponents. Always double-check your work to ensure accuracy and make sure you understand each step. With consistent practice, you’ll master these simplification techniques and be well on your way to becoming an algebra expert!
(d) Simplifying a × b
Finally, let's look at the expression a × b. In this case, we have two different variables, a and b, and there are no exponents or coefficients to deal with. This expression is already in its simplest form. There's nothing more we can do to combine or simplify it further. So, a × b is simply written as ab. This might seem too simple, but it's an important concept to understand. Sometimes, expressions are already in their simplest form, and recognizing this can save you time and effort. Remember that in algebra, different variables cannot be combined unless there's an operation that allows it, such as addition or subtraction with like terms. In this case, a and b are distinct variables, and the only operation between them is multiplication, so we just write them next to each other to indicate multiplication. Keep practicing identifying when an expression is already simplified, and you'll become more efficient in your algebraic manipulations. Always double-check your understanding and make sure you're comfortable with the basic concepts. With consistent practice, you’ll develop a strong foundation in algebra and be able to tackle more complex problems with confidence!
Simplifying algebraic expressions involves breaking down complex expressions into their simplest forms by using the rules of arithmetic and algebra. Whether it’s combining constants, applying exponent rules, or recognizing when an expression is already simplified, these skills are essential for success in mathematics. Remember to always double-check your work and practice consistently to build your confidence and proficiency. Happy simplifying!