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2 years ago

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5. Circle the letter of any card that gives an example of like terms. A. 1\4c and -9c. B. 2.2n and 2.2. C. 6y and 6x. D. 5d^2 an

d d^2. E. 30 and -25.

1 answer:

kolezko [41]2 years ago

6 0

Answer: D

Step-by-step explanation: 5d^2 and d^2 share the same variable

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Prove or disprove (from i=0 to n) sum([2i]^4) <= (4n)^4. If true use induction, else give the smallest value of n that it doe

ddd [48]

Answer:

The statement is true for every n between 0 and 77 and it is false for $n\geq 78$

Step-by-step explanation:

First, observe that, for n=0 and n=1 the statement is true:

For n=0: $\sum^{n}_{i=0} (2i)^4=0 \leq 0=(4n)^4$

For n=1: $\sum^{n}_{i=0} (2i)^4=16 \leq 256=(4n)^4$

From this point we will assume that $n\geq 2$

As we can see, $\sum^{n}_{i=0} (2i)^4=\sum^{n}_{i=0} 16i^4=16\sum^{n}_{i=0} i^4$ and $(4n)^4=256n^4$ . Then,

$\sum^{n}_{i=0} (2i)^4 \leq(4n)^4 \iff \sum^{n}_{i=0} i^4 \leq 16n^4$

Now, we will use the formula for the sum of the first 4th powers:

$\sum^{n}_{i=0} i^4=\frac{n^5}{5} +\frac{n^4}{2} +\frac{n^3}{3}-\frac{n}{30}=\frac{6n^5+15n^4+10n^3-n}{30}$

Therefore:

$\sum^{n}_{i=0} i^4 \leq 16n^4 \iff \frac{6n^5+15n^4+10n^3-n}{30} \leq 16n^4 \\\\ \iff 6n^5+10n^3-n \leq 465n^4 \iff 465n^4-6n^5-10n^3+n\geq 0$

and, because $n \geq 0$ ,

$465n^4-6n^5-10n^3+n\geq 0 \iff n(465n^3-6n^4-10n^2+1)\geq 0 \\\iff 465n^3-6n^4-10n^2+1\geq 0 \iff 465n^3-6n^4-10n^2\geq -1\\\iff n^2(465n-6n^2-10)\geq -1$

Observe that, because $n \geq 2$ and is an integer,

$n^2(465n-6n^2-10)\geq -1 \iff 465n-6n^2-10 \geq 0 \iff n(465-6n) \geq 10\\\iff 465-6n \geq 0 \iff n \leq \frac{465}{6}=\frac{155}{2}=77.5$

In concusion, the statement is true if and only if n is a non negative integer such that $n\leq 77$

So, 78 is the smallest value of n that does not satisfy the inequality.

Note: If you compute $(4n)^4- \sum^{n}_{i=0} (2i)^4$ for 77 and 78 you will obtain:

$(4n)^4- \sum^{n}_{i=0} (2i)^4=53810064$

$(4n)^4- \sum^{n}_{i=0} (2i)^4=-61754992$

7 0

3 years ago

When a certain number is multiplied by 14 and the product is then multiplied by 32, the result is 60. what is the number?

Stolb23 [73]

It might be 0.133928571.....? I am unsure though. Sorry if this doesn't help

7 0

2 years ago

Tell whether the statement is sometimes, always, or never true. Explain your reasoning. The x-coordinate of a point on the x-axi

Tom [10]

I think this would be sometimes true

3 0

3 years ago

Read 2 more answers

What is the slope for this graph?

sergeinik [125]

Answer:

the slope is 1

Step-by-step explanation:

you could just leave it as x in the equation

3 0

2 years ago

Read 2 more answers

I'm confused on how to do this. Please explain step by step. If you respond with just the answer I WILL report you. I would like

Tcecarenko [31]

Answer:

Exact Form:

√2−1

Decimal Form:

0.41421356

…

Step-by-step explanation:

Since 9π/8 is not an angle where the values of the six trigonometric functions are known, try using half-angle identities.

9π/8 is not an exact angle

First, rewrite the angle as the product of 1/2 and an angle where the values of the six trigonometric functions are known. In this case, 9π/8 can be rewritten as

(1/2)* 9π/8 tan ((1/2)* 9π/8)

Use the half-angle identity for tangent to simplify the expression. The formula states that

tan (0/2)=sin(0)/1+cos(0) sin(9π/4)/1+cos(9π/4)

Simplify

Remove full rotations of 2π until the angle is between 0 and 2π.

sin(π/4)/1+cos(9π/4)

The exact value of sin(π/4) is √2/2

√2/2/1+cos(9π/4)

Simplify the Denominator

Remove full rotations of 2π until the angle is between 0 and 2π.√2/2/1+cos(π4)

The exact value of cos(π/4) is √2/2.√2/2/1+√2/2

To write 1/1 as a fraction with a common denominator, multiply by 2/2 .√2/2/1/1⋅2/2+√2/2

Write each expression with a common denominator of 2, by multiplying each by an appropriate factor of 1.

Combine.

√2/2/1⋅2/1⋅2+√2/2

Multiply 2 by 1

√2/2/1⋅2/2+√2/2

Combine the numerators over the common denominator.

√2/2/1⋅2+√2/2

Multiply 2 by 1

√2/2/2+√2/2

Multiply the numerator by the reciprocal of the denominator

√2/2 ⋅ 2/2+√2

Cancel the common factor of 2 .

Factor out the greatest common factor 2

√2/2⋅1 ⋅ 2⋅1/2+√2

Cancel the common factor

√2/2⋅1 ⋅ 2⋅1/2+√2

Rewrite the expression.

√2/1 ⋅ 1/2+√2

Simplify

Multiply √2/1 and 1/2+√2

√2/2+√2

Multiply √2/2+√2 by 2−√2/2−√2

Combine

√2(2−√2)/(2+√2)(2−√2)

Expand the denominator using the FOIL method.

√2(2−√2)/4−2√2+√2⋅2−√2^2

Simplify

√2(2−√2)/2

Apply the distributive property

√2⋅2+√2(−√2)/2

Move 2 to the left of the expression √2⋅2.

2⋅√2+√2(−√2)/2

Simplify

√2(−√2) .

Raise √2 to the power of 1 .

2⋅√2−(√2^1√2)/2

Raise √2 to the power of 1 .

2⋅√2−(√2^1√2^1)/2

Use the power rule a^m a^n=a^m+n to combine exponents.

2⋅√2−√2^1+1/2

Add 1 and 1 .

2⋅√2−√2^2/2

Simplify each term.

Multiply 2 by √2 .

2√2−√2^2/2

Rewrite √2^2 as 2 .

2√2−1⋅2/2

Multiply −1 by 2.

2√2−2/2

Reduce the expression by cancelling the common factors.

Factor 2 out of 2√2.

2(√2)−2/2

Factor 2 out of −2.

2(√2)+2⋅−1/2

Factor 2 out of

2(√2)+2(−1)2(√2−1)/2

Cancel the common factors.

Factor 2 out of 2 .

2(√2−1)/2(1)

Cancel the common factor.

2(√2−1)/2⋅1

Rewrite the expression.

√2−1/1

Divide √2−1 by 1 .

√2−1

The result can be shown in multiple forms.

Exact Form:

√2−1

Decimal Form:

0.41421356…

Hope it help. Good luck.

4 0

3 years ago