// Numbas version: exam_results_page_options {"name": "Simon's copy of Algebra IV: Properties of indices (1) - Multiplication/Division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "name": "Simon's copy of Algebra IV: Properties of indices (1) - Multiplication/Division", "tags": [], "variablesTest": {"condition": "", "maxRuns": 100}, "functions": {}, "extensions": [], "preamble": {"css": "fraction {\n display: inline-block;\n vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n float: left;\n width: 100%;\n text-align: center;\n line-height: 2.5em;\n}\nfraction > numerator {\n border-bottom: 1px solid;\n padding-bottom: 5px;\n}\nfraction > denominator {\n padding-top: 5px;\n}\nfraction input {\n line-height: 1em;\n}\n\nfraction .part {\n margin: 0;\n}\n\n.table-responsive, .fractiontable {\n display:inline-block;\n}\n.fractiontable {\n padding: 0; \n border: 0;\n}\n\n.fractiontable .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n border-bottom: 1px solid black; \n text-align: center;\n}\n\n\n.fractiontable tr {\n height: 3em;\n}", "js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');"}, "advice": "

Recall the laws of indices to help solve the problems:

$x^a \\times x^b = x^{a+b}$

$x^a \\div x^b = x^{a-b}$

$x^{-a} = \\frac{1}{x^a}$

$(x^a)^b = x^{ab}$

$(\\frac{x}{y})^a = \\frac{x^a}{y^a}$

$x^\\frac{a}{b} = (\\sqrt[b]{x})^{a}$

$x^0 = 1$

Worked Solutions:

Part a) $x^{(\\var{a}+\\var{b})}=\\simplify{x^{({a}+{b})}}$

Part b) $p^{(\\var{c}+\\var{d})}=\\simplify{p^{({c}+{d})}}$

Part c) $\\var{a}^\\var{f}\\times{k^{(\\var{b}\\times\\var{f})}}=\\simplify{{a}^{f}*k^{({b}*{f})}}$

Part d) $y^{((\\var{a}+\\var{b})/(\\var{a}\\times\\var{b}))}=y^{\\frac{\\simplify{{a}+{b}}}{\\simplify{{a}*{b}}}}$

Part e) $c^{(\\var{a}-\\var{b})}=c^\\simplify{({a}-{b})}$

Part f) $\\frac{\\var{a}}{\\var{b}}h^{\\var{c}-\\var{d}}=\\frac{\\var{a}}{\\var{b}}{\\simplify{h^{{c}-{d}}}}$

Part g) $\\frac{4^\\var{g}}{2^\\var{h}}\\times{d^{\\var{g}-\\var{h}}}=\\simplify{(4^{g})/(2^{h})*d^{g-h}}$

Part h) $\\frac{6^\\var{g}}{9^\\var{h}}\\times{p^{\\var{h}\\var{j}-\\var{g}\\var{f}}}=\\simplify{(6^{g})/(9^{h})*p^{h*j-g*f}}$

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Simplify each of the following expressions, giving your answer in its simplest form.

Click 'Show steps' for guidance on which index law is applicable.

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$x^\\var{a} \\times x^\\var{b}$

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Use the following indices law to help answer this question:

$x^a \\times x^b = x^{a+b}$

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$p^\\var{c} \\times p^\\var{d}$

Use the following law to help answer this question:

$x^a \\times x^b = x^{a+b}$

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$(\\var{a}k^\\var{b})^\\var{f}$

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Use the following law to answer this question:

$(ax^b)^c = a^cx^{bc}$

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$y^{1/\\var{a}} \\times y^{1/\\var{b}}$

Use the following law:

$x^a \\times x^b = x^{a+b}$

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$c^\\var{a}$$c^\\var{b}$

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Use the following law:

$x^a \\div x^b = x^{a-b}$

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$\\var{a}h^\\var{c}$$\\var{b}h^\\var{d}$

Use the following law to answer this question:

$\\frac{ax^c}{bx^d}= \\frac{a}{b}x^{(c-d)}$

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$(4d)^\\var{g}$$(2d)^\\var{h}$

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This question differs from part f due to the brackets. Using principles of BODMAS, the brackets need to be expanded first.

$(4d)^\\var{g}$ expands to $4^\\var{g}d^\\var{g}$ and $(2d)^\\var{h}$ expands to $2^\\var{h}d^\\var{h}$

Now you are left with a simple division question as follows:

$\\frac{4^{\\var{g}}d^{\\var{g}}}{2^{\\var{h}}d^{\\var{h}}}$

Use the principle:

$\\frac{ax^c}{bx^d}= \\frac{a}{b}x^{(c-d)}$ to answer the question.

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Simplifying indices.

"}, "type": "question", "contributors": [{"name": "Sarah Turner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/881/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Sarah Turner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/881/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}