Math Made Easy: Simplifying Expressions And Solving Problems

Hey guys, let's dive into some cool math problems! We'll break down how to simplify expressions, calculate values, and play around with exponents. Don't worry; it's not as scary as it sounds! We'll take it step by step, so you can follow along easily. Get ready to sharpen your math skills and boost your confidence. This is all about making math fun and understandable, so let's jump right in!

Simplifying Expressions: Your First Math Adventure!

Let's start with simplifying some expressions. This is like taking a complex puzzle and making it simpler to solve. It's all about reducing things to their most basic form. This will help you understand the core concepts and practice the building blocks of more complex math problems. This will lay the foundation for understanding math concepts. Remember, practice makes perfect, so let's start practicing!

1. Write the simplest form of: a. √169 = ? Alright, guys, the square root of 169 is like asking, "What number multiplied by itself equals 169?" Think about it for a sec. We know that 12 x 12 = 144, and 13 x 13 = 169. So, the simplest form of √169 is 13. Easy peasy! b. 7⁷ . 7⁻² + 5⁰ - √4 = ? Okay, this one looks a bit more complex, but we can totally handle it! Let's break it down step by step: * First part: 7⁷ . 7⁻². When multiplying numbers with the same base (in this case, 7), we add the exponents. So, 7⁷ . 7⁻² = 7⁷⁻² = 7⁵. * Second part: 5⁰. Any number raised to the power of 0 equals 1. So, 5⁰ = 1. * Third part: √4. The square root of 4 is 2 (because 2 x 2 = 4). * Putting it all together: We now have 7⁵ + 1 - 2. To calculate 7⁵, we multiply 7 by itself five times (7 x 7 x 7 x 7 x 7), which equals 16,807. So, 16,807 + 1 - 2 = 16,806. Voila! c. ²log 144 + log 1000 = ? This involves logarithms, but let's break it down. ²log 144 means "the power to which 2 must be raised to get 144." This is not straightforward, so we can simplify it using log rules. We'll also use the common log (base 10). * ²log 144: We can rewrite this using the change of base formula: ²log 144 = (log 144) / (log 2). * log 1000: This is the common log (base 10). So, log 1000 = 3 (because 10³ = 1000). * Calculation: To find ²log 144, we first need to find the value of log 144, which is approximately 2.15. Then, we divide that by log 2, which is approximately 0.30. This gives us approximately 7.17. * Final step: Add the results. So, 7.17 + 3 = 10.17. Keep in mind that some values are rounded.

This part is all about understanding the basic concepts. Simplifying these expressions is not just about getting the right answer, it's about understanding the process.

Finding the Value of Expressions: Math Detective Time!

Let's move on to finding the value of expressions. This is where we apply our simplification skills to get a numerical answer. It's like being a math detective, solving a case step by step. Finding the value of expressions is like cracking a code. Let's dive right in!

2. Determine the value of (5 - √18) + √98 = ?

   • Simplify √18: We can break down √18 as √(9 x 2) = √9 x √2 = 3√2.
   • Simplify √98: We can break down √98 as √(49 x 2) = √49 x √2 = 7√2.
   • Rewrite the expression: Now, we have (5 - 3√2) + 7√2.
   • Combine like terms: -3√2 + 7√2 = 4√2.
   • Final step: So, (5 - √18) + √98 = 5 + 4√2. We can't simplify this further to get a single numerical answer because of the square root. The best simplified form is 5 + 4√2.

This is an important step as you start working on more complex mathematical problems.

Simplifying Exponential Forms: Power Up Your Math Skills!

Now, let's simplify exponential forms. This is all about working with exponents and making expressions cleaner and easier to manage. This is essential for understanding various mathematical concepts. Simplifying exponents means we're trying to make the expression more readable and easier to work with. Let's simplify it!

3. Create the simplest exponential form of:

(16 a⁵ b⁷ c⁻²) / (8 a² b⁷ c⁵) = ?

• Numbers: 16 / 8 = 2. * 'a' terms: When dividing exponents with the same base, subtract the exponents. So, a⁵ / a² = a⁵⁻² = a³. * 'b' terms: b⁷ / b⁷ = b⁷⁻⁷ = b⁰ = 1 (because anything to the power of 0 is 1). The 'b' terms effectively cancel out. * 'c' terms: c⁻² / c⁵ = c⁻²⁻⁵ = c⁻⁷. * Putting it all together: The simplified form is 2 a³ c⁻⁷. We can also write this as (2a³) / c⁷, since c⁻⁷ is the same as 1 / c⁷. Always remember that negative exponents mean the term is in the denominator. The result is therefore 2 a³/ c⁷.

This step is important, especially if you plan to do more complicated work with exponents.

Calculating Values: Math in Action!

Let's move on to calculating values. It is the application of all the previous things we've learned. This section is essential for understanding and working with complex mathematical formulas. This allows us to get the final answers to the problems. Let's go!

4. Calculate the value of 2¹²⁷ - 2¹²⁶ = ?

   • Factoring: Notice that both terms have 2 raised to some power. We can factor out the smaller power, 2¹²⁶.
   • Rewrite the expression: 2¹²⁷ - 2¹²⁶ = 2¹²⁶ * 2¹ - 2¹²⁶ * 1.
   • Factor out 2¹²⁶: 2¹²⁶ (2 - 1).
   • Simplify: 2¹²⁶ (1) = 2¹²⁶.

   So, the answer is 2¹²⁶. In this case, it is best to leave it like this; calculating 2¹²⁶ would be a very large number.

Solving Logarithm Problems: Decoding the Logarithms!

Let's solve some logarithm problems. This type of math is super useful in different fields. Remember, learning the basics is important. So let's solve some math problems.

5. If log 3 = a, then ?

The question is incomplete. Since we are given log 3 = a, we need a complete question to solve. The user is missing the question for it. Let's assume the question is: