Simplifying Algebraic Expressions: A Step-by-Step Guide

by TextBrain Team 56 views

Hey guys! Algebra can seem like a beast sometimes, but breaking it down step by step makes it way less scary. Let's tackle these expressions one at a time. I will guide you through simplifying these algebraic expressions. So, grab your pencils, and let's dive in!

1) Simplifying (m³) (-16m²)

When simplifying expressions involving exponents, especially when dealing with the same base, it's all about applying the rules of exponents correctly. In this particular case, we are focusing on the product of powers property. This property states that when you multiply two exponential terms with the same base, you add their exponents. Let's break down the expression (m³) (-16m²) to see how this works.

Our expression is (m³) (-16m²). The first part, m³, means 'm' raised to the power of 3, or m * m * m. The second part, -16m², means -16 multiplied by 'm' raised to the power of 2, or -16 * m * m.

To simplify, we multiply the coefficients (the numbers in front of the variables) and then apply the product of powers rule to the variable 'm'.

The coefficient of m³ is implicitly 1, so we have 1 * -16, which equals -16.

Now, for the variable part, we have m³ * m². According to the product of powers rule, we add the exponents: 3 + 2 = 5. So, m³ * m² = m⁵.

Putting it all together, we get -16m⁵. This is the simplified form of the expression (m³) (-16m²).

Therefore, (m³) (-16m²) simplifies to -16m⁵. This is a straightforward application of the product of powers rule, making the simplification process quite manageable once you understand the rule.

2) Simplifying (-a²x²)(ax³y)

Okay, let's break down how to simplify the algebraic expression (-a²x²)(ax³y). When simplifying expressions like this, we need to multiply the coefficients (the numbers in front of the variables) and then multiply the variables, remembering to add the exponents of like variables. This combines the commutative and associative properties of multiplication along with the product of powers rule for exponents.

Here’s the expression: (-a²x²)(ax³y).

First, let's identify the coefficients and variables:

  • -a²x² can be thought of as -1 * a² * x²
  • ax³y can be thought of as 1 * a * x³ * y

Now, multiply the coefficients: -1 * 1 = -1.

Next, multiply the variables. Remember to add the exponents when multiplying like variables:

  • For 'a', we have a² * a = a^(2+1) = a³
  • For 'x', we have x² * x³ = x^(2+3) = x⁵
  • For 'y', we only have one 'y', so it remains as 'y'

Combine these results: -1 * a³ * x⁵ * y, which simplifies to -a³x⁵y.

So, (-a²x²)(ax³y) simplifies to -a³x⁵y. Remember, the key is to multiply the coefficients and then add the exponents of the like variables.

3) Simplifying (²x²y³) 4 ±½ 5 4

Alright, this expression (²x²y³) 4 ±½ 5 4 looks a bit unusual with those extra numbers hanging around. It seems like there might be some missing operators or a typo. However, assuming we're focusing on the core algebraic part and treating the surrounding numbers as coefficients or separate terms, let’s simplify the central part, which is x²y³.

The expression appears to have extraneous numbers and symbols around the term x²y³, which complicates direct simplification. If the intention is to simplify only the term x²y³, then it is already in its simplest form as the variables x and y are distinct and have no common factors to reduce. So, x²y³ remains as x²y³.

Now, let's consider the entire expression: (²x²y³) 4 ±½ 5 4. If we treat the numbers as coefficients, we might interpret this as a series of operations. However, without clear operators, it's challenging. If we treat them as coefficients, let's try to simplify step by step, assuming the intended expression is something like 2 * x² * y³ * 4 ± ½ * 5 * 4.

  1. Simplifying Coefficients: If we multiply the coefficients together, we have 2 * 4 * 5 * 4 which gives us 160. The ± ½ is ambiguous without context, but let's ignore it for simplicity in this case and only multiply the whole numbers.
  2. Combine with Variables: Now we combine the simplified coefficient with the variable term. So, we have 160 * x² * y³ = 160x²y³.

So, with these assumptions and simplifications, we get 160x²y³.

4) Simplifying (10 ab²c⁴) (1a²bc²)

Now, let's tackle the expression (10 ab²c⁴) (1a²bc²). To simplify this, we'll multiply the coefficients and then multiply the variables, adding the exponents of like variables, just as we did before. This relies on the commutative and associative properties of multiplication, as well as the product of powers rule.

Here’s the expression: (10 ab²c⁴) (1a²bc²).

First, let's identify the coefficients and variables:

  • 10 ab²c⁴ means 10 * a * b² * c⁴
  • 1a²bc² means 1 * a² * b * c²

Multiply the coefficients: 10 * 1 = 10.

Next, multiply the variables, adding the exponents when multiplying like variables:

  • For 'a', we have a * a² = a^(1+2) = a³
  • For 'b', we have b² * b = b^(2+1) = b³
  • For 'c', we have c⁴ * c² = c^(4+2) = c⁶

Combine these results: 10 * a³ * b³ * c⁶, which simplifies to 10a³b³c⁶.

So, (10 ab²c⁴) (1a²bc²) simplifies to 10a³b³c⁶. Remember, the key is to multiply the coefficients and then add the exponents of the like variables.

5) Simplifying (m²n³p²) (4 m²np) · ( — ±² m²n³p)

Okay, let's simplify the expression (m²n³p²) (4 m²np) · ( — ±² m²n³p). This looks a little complex because of the coefficients and the extra symbols. It seems there's a typo with "— ±²", which might be intended as a fractional or negative coefficient. For the sake of simplicity, I will assume that is -2.

Here’s the expression: (m²n³p²) (4 m²np) · (-2 m²n³p).

First, identify the coefficients and variables:

  • m²n³p² means 1 * m² * n³ * p²
  • 4 m²np means 4 * m² * n * p
  • -2 m²n³p means -2 * m² * n³ * p

Multiply the coefficients: 1 * 4 * -2 = -8.

Next, multiply the variables, adding the exponents when multiplying like variables:

  • For 'm', we have m² * m² * m² = m^(2+2+2) = m⁶
  • For 'n', we have n³ * n * n³ = n^(3+1+3) = n⁷
  • For 'p', we have p² * p * p = p^(2+1+1) = p⁴

Combine these results: -8 * m⁶ * n⁷ * p⁴, which simplifies to -8m⁶n⁷p⁴.

So, (m²n³p²) (4 m²np) · (-2 m²n³p) simplifies to -8m⁶n⁷p⁴. Simplifying expressions like this involves multiplying coefficients and summing the exponents of like variables. Be mindful of the signs and make sure each term is accounted for!