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Keep exact values, preserve signs and count sequence steps correctly.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
Simplify numerical expressions and sequence rules while preserving the domain, signs and exact value.
Parentheses determine which quantity an operation acts on. Negative three squared, when written as (-3) squared, is nine because the entire negative number is multiplied by itself. The expression -(3 squared) is negative nine because negation is outside the square. When spoken wording could be unclear, rewrite the expression with explicit grouping. Multiplication and division have equal precedence and are evaluated from left to right unless grouping says otherwise; the same applies to addition and subtraction.
| Term | What it means |
|---|---|
| Reciprocal | For a nonzero number, the number that multiplies it to one. |
| Principal square root | The nonnegative number whose square equals the given nonnegative value. |
| Arithmetic sequence | A sequence with a fixed difference between consecutive terms. |
| Geometric sequence | A sequence with a fixed ratio between consecutive terms. |
Fractions, decimals, powers and roots are different ways to represent quantities. A simplification is useful only if it preserves the original value and any restrictions on variables. Cancelling a factor is valid; cancelling a term from a sum is not. Taking a square root chooses the nonnegative principal root; solving a squared equation may require both signs. Counting the twelfth term of a sequence means making eleven transitions from the first term, not twelve.
These distinctions matter more than the size of the numbers. A large power may simplify immediately by an exponent rule, whereas a small expression can conceal an excluded denominator or a negative sign. Read the structure first. Decide whether the task needs an exact value, an approximation, a comparison or a particular sequence term. The most efficient method often avoids calculating quantities that the question never requests.
Another way: steps
Read the structure and restrictions; choose a valid representation; simplify exactly; check signs and index boundaries.
To add fractions, express them with a common denominator. Three eighths plus one sixth becomes nine twenty-fourths plus four twenty-fourths, or thirteen twenty-fourths. Adding denominators would change the size of the units being counted. For multiplication, multiply numerators and denominators, cancelling common factors before expanding if that makes the work simpler. For division by a nonzero fraction, multiply by its reciprocal. A denominator of zero is never permitted. If an expression contains a variable denominator, retain that restriction even if later cancellation hides the denominator.
Multiplying 0.004 by 3000 gives twelve, not 0.012 or 12000. Expressing the quantities as 4 times ten to the negative third and 3 times ten to the third makes the cancellation visible. Scientific notation separates a coefficient from its power-of-ten scale. Estimation can catch a misplaced decimal before exact work: 0.49 times 202 is close to one half of two hundred, or one hundred. If your result is near one or ten thousand, revisit the scale rather than accepting the calculator display without interpretation.
For the same nonzero base, multiplication adds exponents and division subtracts them. Raising a power to another integer power multiplies exponents. A negative exponent denotes a reciprocal, not a negative value: 2 to the negative fourth is one sixteenth. Any nonzero base to exponent zero is one. Do not extend a multiplication rule to addition: 2 cubed plus 2 squared is twelve, not 2 to the fifth power. Keep the base fixed when applying a rule; powers with different bases do not combine just because their exponents look similar.
The principal square root of forty-nine is seven. The equation x squared equals forty-nine has two real solutions, seven and negative seven. Thus the square root of x squared equals the absolute value of x, not always x. If x is known nonnegative, the absolute value can be replaced by x. If the sign is unknown, retain it or split into cases. A square root distributes over a product of nonnegative quantities but not over an arbitrary sum: the square root of nine plus sixteen is five, whereas three plus four is seven.
The absolute value of x is its distance from zero, so it cannot be negative. The distance between a and b is the absolute value of a - b, which is the same as the absolute value of b - a. An equation |x - 4| = 3 describes points three units from four and therefore has solutions one and seven. An inequality |x - 4| < 3 describes the open interval between them. Connect the symbols to a number line before manipulating cases; the geometric meaning often prevents an omitted solution or an incorrect endpoint.
A few listed numbers alone do not logically determine every later term. In these questions the rule is stated: add a fixed amount, multiply by a fixed factor, or use an explicit recurrence. For an arithmetic sequence starting at a with difference d, term n is a + (n - 1)d. For a geometric sequence starting at a with ratio r, term n is a times r to power n - 1. Check n = 1 in either formula; it must return the first term. This quick boundary test catches a common off-by-one error.
For positive bases above one, increasing a positive integer exponent increases the power. For a base strictly between zero and one, increasing the exponent decreases it. A negative base requires attention to odd and even exponents because the sign alternates. When comparing powers, try rewriting them with a common base or exponent. For example, 8 squared and 4 cubed both equal 2 to the sixth power. This structural equivalence establishes equality without calculating sixty-four twice. Restrictions determine which monotonicity claims are valid. If a denominator contains a power of a variable, explicitly exclude zero before applying cancellation or negative-exponent rules; a simplified expression does not restore excluded inputs.
A laboratory may describe a culture as multiplying by a fixed factor each hour. That is a geometric model: after four complete hourly intervals, apply the multiplier four times to the initial amount. If the initial count is described as the first recorded observation rather than the count at time zero, check which observation the question requests. Time intervals and observation indices can differ by one even though the physical story is the same.
The mathematical model does not assert that a real culture can grow indefinitely without resource limits. Within a stated practice problem, use the supplied constant-growth assumption for the specified period. Outside that interval, the conclusion may not apply. This is a useful distinction between solving a model and proving that the model describes every real situation.
Before using a calculator, estimate the order of magnitude and write the exponent. After computing, check that growth with a factor above one increased the amount and that you used the correct number of intervals. A wrong exponent is a modeling error; extra decimal precision cannot repair it.
A negative exponent is a reciprocal, not a negative sign. The principal square root is nonnegative, but a squared equation can have two solutions. An exponent rule for products is not a rule for sums. Cancelling a factor does not cancel an arbitrary term. A sequence formula must reproduce its stated first term.
Identify the entire divisor.
Compute (7/10) divided by (14/15).
The divisor is a nonzero fraction.
Multiply by its reciprocal.
(7/10) times (15/14).
Division by a quantity reverses its multiplicative effect.
Cancel common factors.
7/14 = 1/2 and 15/10 = 3/2.
Cancellation acts on factors of the product.
Multiply and check magnitude.
The result is 3/4 = 0.75.
Dividing 0.7 by a number just below one should increase it slightly.
Read the variable restriction.
x = -6; evaluate the square root of x squared, minus x.
The subtraction is outside the square root.
Compute the square.
x squared = 36.
Squaring removes the sign at this intermediate stage.
Take the principal square root.
The square root of 36 is 6.
The principal root is nonnegative.
Substitute into the full expression.
6 - (-6) = 12.
Subtracting x adds six because x is negative.
Reject the invalid shortcut.
Replacing the root by x would incorrectly give zero.
The correct identity is the square root of x squared equals |x|.
Read the starting point and rule.
The first term is 5; each later term triples.
Find the first term greater than one thousand.
Write the general term.
Term n is 5 times 3 to power n - 1.
There are n - 1 transitions from the first term.
Remove the common scale.
We need 3 to power n - 1 greater than 200.
Dividing by positive five preserves the inequality.
Locate neighboring powers.
3 to the fourth is 81; 3 to the fifth is 243.
The first exceeds neither; the second exceeds two hundred.
Recover the term index.
n - 1 = 5, so n = 6.
The exponent counts transitions, not the term number.
Verify the first crossing.
Term five is 405 and term six is 1215.
The prior term is below the threshold, so term six is the first qualifying one.
Identify the initial term and difference.
First term -8; common difference 5; find term 10.
Count transitions from the first term.
Write the expression before calculating.
-8 + (10 - 1) times 5.
Decide why there are nine additions rather than ten.
Calculate and check a neighboring term.
What is 3/8 + 1/4? Enter the exact decimal.
What is 2 to the power 7 divided by 2 to the power 4?
What is (3/4) divided by (5/8)? Enter the exact decimal.
What is 0.006 times 4000?
What is 5 to the power zero plus 2 to the power negative three? Enter the exact decimal.
What is the nonnegative square root of 144/25? Enter the exact decimal.
A sequence has first term 7 and each subsequent term is 4 greater than the preceding term. What is its twelfth term?
A sequence begins 3, 6, 12, 24 and continues by multiplying each term by 2. What is its eighth term?
What is the distance on the number line between -3.5 and 2.25?
If x is negative and x squared equals 81, what is x cubed?
A culture starts with 500 cells and triples every complete hour. Under this model, how many cells are present after 4 hours?
Answer three reserved questions without hints. These cases are separate from lesson practice.
A sequence has first term 20 and each subsequent term is 3 less than the preceding term. What is its ninth term?
If a is positive and a squared equals 0.04, what is 1/a?
What is (5/6 - 1/4) multiplied by 12?
Tomorrow, contrast the square root of x squared with x, then find a term in an arithmetic sequence and one in a geometric sequence. Explain why the exponent is one less than the term number when the first term is given.
17. Complete an arithmetic-sequence calculation, step 3
Term ten is 37; term nine is 32.
Their difference is five, consistent with the stated rule.