Consider the algebraic expression (2x y)(a-7b)^3

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Well, well, well, looks like we have a party going on here with our algebraic expression! Let's break it down and have some fun!

First, we have (2xy), our cool gang of variables. They are always up for a good time.

Next, we have (a-7b)^3, our mischievous trio of (a-7b) being cubed. They love to raise the roof, or in this case, raise each other to the power of 3!

Now, let's combine our party animals by multiplying them together. It's like a dance-off of variables!

(2xy)(a-7b)^3 = 2xy * (a-7b)^3

Oh boy, we have an explosive combination! That means we need to apply our multiplication powers carefully.

When multiplying (2xy) with (a-7b)^3, we distribute the 2xy to each term inside the parentheses:

= 2xy * (a^3 - 3 * a^2 * 7b + 3 * a * (7b)^2 - (7b)^3)

Look at that! We've used our multiplication skills like party tricks. Let's simplify our expression further.

= 2xy * (a^3 - 21a^2b + 3a * 49b^2 - 343b^3)

Now we have a grand finale! Our algebraic expression is all simplified and ready for some serious mathematical fun!

So, my friend, the simplified expression is 2xy(a^3 - 21a^2b + 147ab^2 - 343b^3). Let's give it a round of applause for being such an entertaining equation!

To expand the algebraic expression (2xy)(a-7b)^3, you can use the binomial expansion formula. By raising the binomial (a-7b) to the third power, you'll get a sum of terms. The formula for expanding a binomial is:

(a-b)^n = C(n,0)(a^n)(b^0) + C(n,1)(a^(n-1))(b^1) + C(n,2)(a^(n-2))(b^2) + ... + C(n,n)(a^0)(b^n)

where C(n, k) = n! / (k!(n - k)!) is the binomial coefficient.

Applying this formula to (a-7b)^3, you can expand it:

(a-7b)^3 = C(3,0)(a^3)(-7b)^0 + C(3,1)(a^2)(-7b)^1 + C(3,2)(a^1)(-7b)^2 + C(3,3)(a^0)(-7b)^3

Now let's simplify each term:

Term 1:

C(3,0)(a^3)(-7b)^0
= 1(1)(-7b)^0
= a^3

Term 2:

C(3,1)(a^2)(-7b)^1
= 3(1)(a^2)(-7b)^1
= -21ab

Term 3:

C(3,2)(a^1)(-7b)^2
= 3(2)(a^1)(-7b)^2
= 42a(-7b)^2
= 42a(49b^2)
= 2058ab^2

Term 4:

C(3,3)(a^0)(-7b)^3
= 1(1)(-7b)^3
= -343b^3

Summing up all the terms, you get the expanded form of the expression (2xy)(a-7b)^3 as:

(2xy)(a-7b)^3 = 2xy(a^3 - 21ab + 2058ab^2 - 343b^3)

To expand the algebraic expression (2xy)(a-7b)^3, you can use the binomial theorem. The binomial theorem states that for any two terms raised to an exponent, the expanded form can be determined using combinations and the powers of the two terms.

Let's go step-by-step to expand the expression:

Step 1: Expand the first term, (2xy).

The term (2xy) is already expanded, so we don't need to do anything with it.

Step 2: Expand the second term, (a-7b)^3.

To expand the binomial (a-7b) raised to the power of 3, we need to apply the binomial theorem.

The binomial theorem states that the coefficient of each term in the expansion can be calculated using combinations. The exponent of the first term decreases by one with each term, while the exponent of the second term increases by one.

Using the formula for combinations (nCr), where n is the exponent and r is the term number (starting from 0), we can expand the binomial (a-7b)^3 as follows:

Term 1: (a^3)*(1)(-7b)^0
Term 2: (a^2)*(-7b)^1*(3C1)
Term 3: (a^1)*(-7b)^2*(3C2)
Term 4: (a^0)*(-7b)^3*(3C3)

Simplifying each term, we get:

Term 1: (a^3)*(1)*(1)
Term 2: (a^2)*(-7b)*(3)
Term 3: (a^1)*(49b^2)*(3)
Term 4: (1)*(-343b^3)*(1)

Combining like terms, we get:

(a^3) + (-21a^2b) + (147ab^2) + (-343b^3)

Step 3: Multiply the expanded terms.

Now that we have expanded both terms, we can multiply them together:

(2xy) * [(a^3) + (-21a^2b) + (147ab^2) + (-343b^3)]

Multiplying each term of (2xy) by each term of the expanded (a-7b)^3, we get:

2xy * (a^3) + 2xy * (-21a^2b) + 2xy * (147ab^2) + 2xy * (-343b^3)

Simplifying each term, we get:

2xya^3 - 42xya^2b + 294xyab^2 - 686xyb^3

So, the expanded form of the expression (2xy)(a-7b)^3 is 2xya^3 - 42xya^2b + 294xyab^2 - 686xyb^3.